From brenti(at-sign)mat.utovrm.it Fri Jan 9 11:14:43 1998 Return-Path: Received: from axp.mat.utovrm.it (axp.mat.utovrm.it [160.80.10.11]) by math.mit.edu (8.8.7/8.8.7) with SMTP id LAA21390 for ; Fri, 9 Jan 1998 11:14:42 -0500 (EST) Received: by axp.mat.utovrm.it (5.65v3.2/1.1.10.5/23Dec96-1228PM) id AA28078; Fri, 9 Jan 1998 17:16:21 +0100 Date: Fri, 9 Jan 1998 17:16:21 +0100 From: Francesco Brenti Message-Id: <9801091616.AA28078(at-sign)axp.mat.utovrm.it> Apparently-To: combinatorics(at-sign)math.mit.edu Dear colleague, I have accepted a (tenured) position at the Associate Professor level al the University of Rome. Therefore, my "coordinates" have changed. The new ones are: Dipartimento di Matematica Universita' di Roma "Tor Vergata" Via della Ricerca Scientifica, 1 00133, Roma, Italy Tel. (39) (6) 7259 4671 Fax. (39) (6) 7259 4699 e-mail: brenti(at-sign)mat.utovrm.it Sincerely yours, Francesco Brenti From bergeron(at-sign)mathstat.yorku.ca Fri Jan 9 12:14:03 1998 Return-Path: Received: from sungod.ccs.yorku.ca (rOyjDlx6KGsuq+WlNKBwI4gUiBJO7sgK(at-sign)sungod.ccs.yorku.ca [130.63.236.104]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id MAA22927; Fri, 9 Jan 1998 12:14:02 -0500 (EST) Received: from [130.63.218.86] (msfac1.math.yorku.ca [130.63.218.86]) by sungod.ccs.yorku.ca (8.8.7/8.6.11) with SMTP id LAA11942; Fri, 9 Jan 1998 11:51:38 -0500 (EST) X-Sender: bergeron(at-sign)pascal.math.yorku.ca Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Fri, 9 Jan 1998 12:01:20 -0400 To: bergeron(at-sign)mathstat.yorku.ca From: bergeron(at-sign)mathstat.yorku.ca (Nantel Bergeron) Subject: FPSAC98 News Update for FPSAC98 OnLine Registration: Visit http://www.math.yorku.ca/bergeron THE CALL FOR PAPERS AND POSTERS IS OVER. CALL FOR SOFTWARE DEMONSTRATION: DEAD LINE Jan 21. PARTICIPANT SUPPORT: DEAD LINE Jan 21. ----------------------------------------------------------------------- 10-th international Conference on Formal Power Series and Algebraic Combinatorics Sunday, June 14 - Friday, June 19, 1998 Fields Institute, Toronto TOPICS: Algebraic and bijective combinatorics and their relations with other parts of mathematics, computer science and physics. CONFERENCE PROGRAM: Invited lectures, contributed presentations, poster sessions and software demonstrations. OFFICIAL LANGUAGES: English and French. INVITED TALKS: G. Benkart (USA) P. Cameron (England) (not confirmed) P. Dehornoy (France) B. Derrida (France) P. Diaconis (USA) C. Godsil (Canada) K. Ono (USA) J. Y. Thibon (France) B. Sturmfels (USA) CALL FOR SOFTWARE DEMONSTRATION: Demonstrations of software relevant to the topics of the conference are encouraged. People interested in giving a software demonstration should submit a short description of the software, including the hardware requirements, before January 21, 1998, by email to fpsac98(at-sign)fields.utoronto.ca PROGRAM COMMITTEE: I. Goulden , Chairman (U. of Waterloo), N. Bergeron (York U.), S. Billey (USA), F. Brenti (Italy), R. Cori (France), S. Dulucq (France), K. Eriksson (Sweeden), O. Foda (Australia), S. Fomin (USA/Russia), I. Gessel (USA), C. Greene (USA), A. Hamel (New Zeland), D. Kim (Korea), C. Krattenthaler (Austria), D. Krob (France), M. Noy (Spain), V. Reiner (USA), C. Reutenauer (UQAM), F. Sottile (U. Toronto), T. Visentin (U. Winnipeg). M. Wachs (USA), H. Yamada (Japan), G. Ziegler (Germany). PARTICIPANT SUPPORT: Limited funds are available for partial support of participants. Requests should contain a letter of recommendation and include the estimated transportation and living expenses as well as the amount of support available from other sources. All requests should be sent in duplicate by January 21, 1998 to the Fields Institute. fpsac98(at-sign)fields.utoronto.ca LOCATION: The conference will take place at the Fields Institute in Toronto, Canada. Registration and afternoon reception will be on Sunday, June 14 from 2:00 - 4:00 p.m. The first talk is scheduled on June 15, 1998 at 9:00 a.m. REGISTRATION: Until April 15, 1998, the registration fee is 200 $CAN and 100 $CAN is offered for students (with verification, such as a letter from the advisor). After April 15, 1998 fees will be 300 $CAN and 150 $CAN, respectively. The registration fee covers all lectures and presentations, program materials, refreshment breaks, a copy of the abstract package, and a dinner on Thursday night. To register Visit http://bergeron(at-sign)mathstat.yorku.ca or contact FPSAC/SFCA 98 Fields Institute 222 College Street Toronto, ON M5T 3J1 Canada fpsac98(at-sign)fields.utoronto.ca web-site: www.fields.utoronto.ca FOR MORE INFORMATION: Contact bergeron(at-sign)mathstat.yorku.ca . For information on Fields Institute: visit www.fields.utoronto.ca ORGANIZATION: N. Bergeron, Chairman (York U.), M. Delest (U. de Bordeaux), F. Sottile (U. Toronto), W. W hiteley (York U.). SPONSOR: Our sponsor are: Fields Institute of Toronto, Centre de Recherches Mathematiques a Montreal, University of Toronto (Connaught fund) and York University . We also have some technical support from Combinatorics and Optimization at Waterloo University. Nantel Bergeron bergeron(at-sign)mathstat.yorku.ca Associate Prof. Mathematics nantel(at-sign)math.harvard.edu York University nantel(at-sign)lacim.uqam.ca http://www.math.yorku.ca/bergeron From bergeron(at-sign)mathstat.yorku.ca Fri Jan 9 12:14:01 1998 Return-Path: Received: from sungod.ccs.yorku.ca (+yP6Tv7LS8NBVmAr2WTmQdJ7sn49wiQU(at-sign)sungod.ccs.yorku.ca [130.63.236.104]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id MAA22925; Fri, 9 Jan 1998 12:14:00 -0500 (EST) Received: from [130.63.218.86] (msfac1.math.yorku.ca [130.63.218.86]) by sungod.ccs.yorku.ca (8.8.7/8.6.11) with SMTP id LAA12725; Fri, 9 Jan 1998 11:59:36 -0500 (EST) X-Sender: bergeron(at-sign)pascal.math.yorku.ca Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Date: Fri, 9 Jan 1998 12:06:37 -0400 To: bergeron(at-sign)mathstat.yorku.ca From: bergeron(at-sign)mathstat.yorku.ca (Nantel Bergeron) Subject: SCFA98 SFCA98: Inscription en ligne: Visitez http://www.math.yorku.ca/bergeron. Les SOUMISSIONS DE COMMUNICATIONS ET DE POSTERS sont termines. SOUMISSIONS DE DEMONSTRATIONS DE LOGICIELS: DATE LIMITE 21 jan. =46INANCEMENT DE DEPLACEMENTS: DATE LIMITE 21 jan. ----------------------------------------------------------------------- 10ieme conference sur les Series Formelles et la Combinatoire Algebriques 14-19 juin 1998, =46ields Institute, Toronto THEMES: Les themes traditionnels du colloque sont la combinatoire algebrique = et bijective dans leurs liens avec les mathematiques, l'informatique et la physique. DEROULEMENT DE LA CONFERENCE: Le colloque comprend des conferences invitees de 60 minutes, des communications selectionnees de 30 minutes, une seance d'affichage de posters selectionnes et des demonstrations de logiciel= s. LANGUES OFFICIELLES: l'anglais et le francais. CONFERENCIER INVITES: G. Benkart (USA) P. Cameron (England) (non confirme) P. Dehornoy (France) B. Derrida (France) P. Diaconis (USA) C. Godsil (Canada) K. Ono (USA) J. Y. Thibon (France) B. Sturmfels (USA) DEMONSTRATIONS DE LOGICIELS: Les demonstrations de logiciels lies aux themes du colloque sont vivement encouragees. Les personnes souhaitant faire une demonstration de logiciel sont invitees a envoyer une courte descript= ion de leur logiciel, precisant les supports materiels necessaires, avant le 21Janvier 1998, par email a fpsac98(at-sign)fields.utoronto.ca COMITE DE PROGRAMME: I. Goulden , Chairman (U. of Waterloo), N. Bergeron (York U.), S. Billey (USA), F. Brenti (Italy), R. Cori (France), S. Dulucq (France), K. Eriksson (Sweeden), O. Foda (Australia), S. Fomin (USA/Russia), I. Gessel (USA), C. Greene (USA), A. Hamel (New Zeland), D. Kim (Korea), C. Krattenthaler (Austria), D. Krob (France), M. Noy (Spain), V. Reiner (USA), C. Reutenauer (UQAM), F. Sottile (U. Toronto), T. Visentin (U. Winnipeg). M. Wachs (USA), H. Yamada (Japan), G. Ziegler (Germany). =46INANCEMENT DE DEPLACEMENTS: Un nombre limite de subventions sera disponible pour pouvoir particip= er au colloque. Les demandes de subvention devront contenir l'avis d'une personnalite scientifique et inclure les renseignements suivants: montant des frais de transport et de sejour, montant des autres sources de financement. Toute demande devra parven= ir en double exemplaire avant le 21Janvier 1998 a fpsac98(at-sign)fields.utoronto.ca. LIEU: La conf=E9rence se d=E9roulera =E0 Fields Institute de Toronto,= Canada. Les inscriptions et une petite r=E9ception auront lieu le dimance 14 juin de 14h00 =E0 16h00. La premi=E8re conf=E9rence est pr=E9vue= pour le 15 juin 1998 =E0 9 heures. INSCRIPTION: Avant le 15 avril 1998, les frais d'inscription sont fixes a $200 (Ca= n) Un tarif reduit de $100 est prevu pour les etudiants (a justifier par une lettre). Pour les demandes d'inscription parvenant apres le 30 Avril 1998, ces frais sont portes a 50% de plus. Les frais l'inscription couvrent les couts pour les instructions, les documents de programme, les rafraichissements pendant les pauses, une copie des actes du colloque, et la reception du jeudi soi= r. Pour s'inscrire, remplir le formulaire SUIVANT ou contacter Fields Institute: FPSAC/SFCA 98 Fields Institute 222 College Street Toronto, ON M5T 3J1 Canada fpsac98(at-sign)fields.utoronto.ca web-site: www.fields.utoronto.ca RESEIGNEMENTS SUPLEMENTAIRES: Pour toute demande de renseignement supplementaire, n'hesitez pas a ecrire a fpsac98(at-sign)fields.utoronto.ca Pour plus d'information sur l'institue Fields, les directions et Toronto, www.fields.utoronto.ca COMITE D"ORGANIZATION: N. Bergeron, Chairman (York U.), M. Delest (U. de Bordeaux), F. Sotti= le (U. Toronto), W. W hiteley (York U.). COMMANDITAIRES Nos commanditaire sont: le Fields Institute de Toronto, le Centre de Recherches Mathematiques a Montreal, l'Universite de Toronto (visitez le departement de mathematiques ) et l'universite York (visitez le departement de mathematiques ). Nous avons aussi le soutient techniques du departement de Combinatoires et d'Optimization a l'universite de Waterloo. Nantel Bergeron bergeron(at-sign)mathstat.yorku.ca Associate Prof. Mathematics nantel(at-sign)math.harvard.edu York University nantel(at-sign)lacim.uqam.ca http://www.math.yorku.ca/bergeron From propp(at-sign)math.mit.edu Wed Jan 14 22:59:52 1998 Return-Path: Received: from pfaff.mit.edu (PFAFF.MIT.EDU [18.87.0.183]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id WAA03638 for ; Wed, 14 Jan 1998 22:59:52 -0500 (EST) Received: (from propp(at-sign)localhost) by pfaff.mit.edu (8.8.7/8.6.9) id WAA14625 for combinatorics(at-sign)math.mit.edu; Wed, 14 Jan 1998 22:59:44 -0500 (EST) Date: Wed, 14 Jan 1998 22:59:44 -0500 (EST) From: Jim Propp Message-Id: <199801150359.WAA14625(at-sign)pfaff.mit.edu> To: combinatorics(at-sign)math.mit.edu Subject: alternating sign matrices I'll be giving a talk on Wednesday, January 21 from 1:00 to 2:30 p.m. in room 2-190 at M.I.T. on the history of the alternating sign matrix formula. It'll be directed at undergraduates, and a bit on the light side (as you might guess from the mock-sensationalistic title of the lecture --- "World's Hardest Counting Problem: Solved!"). Jim Propp From propp(at-sign)math.mit.edu Thu Jan 22 09:03:08 1998 Return-Path: Received: from pfaff.mit.edu (PFAFF.MIT.EDU [18.87.0.183]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id JAA20004 for ; Thu, 22 Jan 1998 09:03:08 -0500 (EST) Received: (from propp(at-sign)localhost) by pfaff.mit.edu (8.8.7/8.6.9) id JAA13051 for combinatorics(at-sign)math.mit.edu; Thu, 22 Jan 1998 09:03:07 -0500 (EST) Date: Thu, 22 Jan 1998 09:03:07 -0500 (EST) From: Jim Propp Message-Id: <199801221403.JAA13051(at-sign)pfaff.mit.edu> To: combinatorics(at-sign)math.mit.edu Subject: MIT Combinatorics Seminar There are still several openings for the MIT Combinatorics Seminar in February. If you would like to speak, please send me, by this weekend, the _title of_ your talk, an _abstract_, and the _dates_ on which you could give your talk (Wednesdays and Fridays, usual time). If we can't accomodate you in February, we should be able to schedule you in March; the spring schedule is still quite open. Jim Propp From kcollins(at-sign)mail.wesleyan.edu Wed Jan 28 10:01:19 1998 Return-Path: Received: from mail.wesleyan.edu (dns.wesleyan.edu [129.133.12.10]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id KAA05009; Wed, 28 Jan 1998 10:01:18 -0500 (EST) Received: from [129.133.253.104] (ppp4.subnet253.wesleyan.edu [129.133.253.104]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id JAA00176; Wed, 28 Jan 1998 09:50:52 -0500 (EST) Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 28 Jan 1998 10:00:13 -0500 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: first announcement Come to the Twenty-eighth one day conference on Combinatorics and Graph Theory Saturday, February 21, 1997 10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063 Schedule 10:00 Katalin Vesztergombi (Yale University) Properties of Distance-Graphs 11:10 Sheila Sundaram (Danbury, Connecticut) Homology of Graph Complexes and Partitions with Forbidden Block Sizes 12:10 Lunch 2:00 Emily H. Moore (Grinnell College and Mt. Holyoke College) Extending Graph Colorings 3:10 Dana Randall (Georgia Institute of Technology) TBA The conferences are supported by an NSF grant which allows us to provide a modest transportation allowance to those attendees who are not local. We also gratefully acknowledge support from Smith College and Wesleyan University. Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)math.smith.edu Karen Collins (Wesleyan University), (860) 685-2169, kcollins(at-sign)wesleyan.edu Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)math.smith.edu From sara(at-sign)math.mit.edu Tue Feb 3 13:12:27 1998 Return-Path: Received: from schubert.mit.edu (SCHUBERT.MIT.EDU [18.87.0.16]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA03461 for ; Tue, 3 Feb 1998 13:12:27 -0500 (EST) Received: (from sara(at-sign)localhost) by schubert.mit.edu (8.8.7/8.8.7) id NAA22367; Tue, 3 Feb 1998 13:12:25 -0500 (EST) Date: Tue, 3 Feb 1998 13:12:25 -0500 (EST) Message-Id: <199802031812.NAA22367(at-sign)schubert.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar X-URL: http://www-math.mit.edu/~sara Welcome back from winter break. The combinatorics seminar will resume this Friday with a talk by Serkan Hosten entitled "The order dimension of the complete graph and Scarf complexes". Abstracts and titles are available on our web page http://www-math.mit.edu/~combin We are still looking for speakers for March, April and May. Please let us know if you would like to give a talk and/or if you have visitors in town who might like to speak. FEBRUARY SCHEDULE: February 6: Serkan Hosten The order dimension of the complete graph and Scarf complexes February 11: Alexander Postnikov Algebras of Chern forms on flag manifolds and forests February 18: Harald Helfgott TBA. February 20: Eva Feitchner On the cohomology algebras of complex subspace arrangements February 25: Richard Stanley Enumerating solutions to equations in finite groups February 27: Alex Burstein Enumeration of words with forbidden patterns ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From propp(at-sign)math.mit.edu Tue Feb 3 21:55:02 1998 Return-Path: Received: from pfaff.mit.edu (PFAFF.MIT.EDU [18.87.0.183]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id VAA18225 for ; Tue, 3 Feb 1998 21:55:01 -0500 (EST) Received: (from propp(at-sign)localhost) by pfaff.mit.edu (8.8.7/8.6.9) id VAA03898 for combinatorics(at-sign)math.mit.edu; Tue, 3 Feb 1998 21:55:02 -0500 (EST) Date: Tue, 3 Feb 1998 21:55:02 -0500 (EST) From: Jim Propp Message-Id: <199802040255.VAA03898(at-sign)pfaff.mit.edu> To: combinatorics(at-sign)math.mit.edu Subject: February schedule Here is a more complete version of the February schedule: FEBRUARY SCHEDULE: February 6: Serkan Hosten The order dimension of the complete graph and Scarf complexes February 11: Alexander Postnikov Algebras of Chern forms on flag manifolds and forests February 18: Harald Helfgott Recent developments in enumeration of tilings February 20: Eva Feitchner On the cohomology algebras of complex subspace arrangements February 25: Richard Stanley Enumerating solutions to equations in finite groups February 27: Alex Burstein Enumeration of words with forbidden patterns From kcollins(at-sign)mail.wesleyan.edu Thu Feb 5 11:36:11 1998 Return-Path: Received: from mail.wesleyan.edu (dns.wesleyan.edu [129.133.12.10]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id LAA27987; Thu, 5 Feb 1998 11:36:11 -0500 (EST) Received: from [129.133.252.114] (ppp14.subnet252.wesleyan.edu [129.133.252.114]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id LAA28798; Thu, 5 Feb 1998 11:23:05 -0500 (EST) Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Thu, 5 Feb 1998 11:34:57 -0500 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: second announcement Come to the Twenty-eighth one day conference on Combinatorics and Graph Theory Saturday, February 21, 1997 10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063 Schedule 10:00 Katalin Vesztergombi (Yale University) Properties of Distance-Graphs 11:10 Sheila Sundaram (Danbury, Connecticut) Homology of Graph Complexes and Partitions with Forbidden Block Sizes 12:10 Lunch 2:00 Emily H. Moore (Grinnell College and Mt. Holyoke College) Extending Graph Colorings 3:10 Dana Randall (Georgia Institute of Technology) Two Dimensional Tilings in Two and Three Dimensions The conferences are supported by an NSF grant which allows us to provide a modest transportation allowance to those attendees who are not local. We also gratefully acknowledge support from Smith College and Wesleyan University. Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)math.smith.edu Karen Collins (Wesleyan University), (860) 685-2169, kcollins(at-sign)wesleyan.edu Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)math.smith.edu From sara(at-sign)math.mit.edu Fri Feb 6 15:58:20 1998 Return-Path: Received: from schubert.mit.edu (SCHUBERT.MIT.EDU [18.87.0.16]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id PAA16738 for ; Fri, 6 Feb 1998 15:58:20 -0500 (EST) Received: (from sara(at-sign)localhost) by schubert.mit.edu (8.8.7/8.8.7) id PAA20172; Fri, 6 Feb 1998 15:58:19 -0500 (EST) Date: Fri, 6 Feb 1998 15:58:19 -0500 (EST) Message-Id: <199802062058.PAA20172(at-sign)schubert.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Date: February 6, 1998 Speaker: Serkan Hosten Title: The order dimension of the complete graph and Scarf complexes Abstract: In this talk we will show how one determines the order dimension (of the poset) of the complete graph on n vertices. The order dimension of a poset is the least number of linear extensions whose intersection realizes the poset. The order dimension of the complete graph was not determined exactly for large number of vertices, and only rough extimates existed. We give a precise description. This result has implications in the combinatorics of Scarf complexes defined by Bayer, Peeva, and Sturmfels to compute free resolutions of monomial ideals. This is joint work with Walter Morris. Upcoming Events: February 11: Alexander Postnikov Algebras of Chern forms on flag manifolds and forests February 18: Harald Helfgott TBA. February 20: Eva Feitchner On the cohomology algebras of complex subspace arrangements February 25: Richard Stanley Enumerating solutions to equations in finite groups February 27: Alex Burstein Enumeration of words with forbidden patterns ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From IARROBIN(at-sign)neu.edu Fri Feb 6 23:26:02 1998 Return-Path: Received: from NUHUB.DAC.NEU.EDU (nuhub.dac.neu.edu [129.10.1.6]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id XAA23868 for ; Fri, 6 Feb 1998 23:26:01 -0500 (EST) From: IARROBIN(at-sign)neu.edu Received: from neu.edu by neu.edu (PMDF V5.1-8 #24746) id <01IT9ZJCBL748X7ZVO(at-sign)neu.edu> for combinatorics(at-sign)math.mit.edu; Fri, 6 Feb 1998 23:25:57 EST Date: Fri, 06 Feb 1998 23:25:56 -0500 (EST) Subject: of possible interest - Seminar at NU Mon Feb 9 To: combinatorics(at-sign)math.mit.edu Message-id: <01IT9ZJCCE4I8X7ZVO(at-sign)neu.edu> X-VMS-To: IN%"combinatorics(at-sign)math.mit.edu" MIME-version: 1.0 Northeastern Univ. GAS seminar, Monday Feb 9 at 1:30 PM at 509 Lake Hall: Vic Reiner (Minnesota): "Resolutions and the homology of chessboard and matching complexes" Abstract: [e use We use Lascoux's resolution for 2 x 2 minors, and some generalizations of it, to calculate the homology of these simplicial complexes - AI From sara(at-sign)math.mit.edu Mon Feb 9 10:38:07 1998 Return-Path: Received: from schubert.mit.edu (SCHUBERT.MIT.EDU [18.87.0.16]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id KAA01896 for ; Mon, 9 Feb 1998 10:38:07 -0500 (EST) Received: (from sara(at-sign)localhost) by schubert.mit.edu (8.8.7/8.8.7) id KAA06433; Mon, 9 Feb 1998 10:38:00 -0500 (EST) Date: Mon, 9 Feb 1998 10:38:00 -0500 (EST) Message-Id: <199802091538.KAA06433(at-sign)schubert.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: [daisymae(at-sign)math.mit.edu: APPLIED MATHEMATICS COLLOQUIUM -- Monday, February 9, 1998] X-URL: http://www-math.mit.edu/~sara Reminder, Vic Reiner will be speaking today in the applied math colloquium on a topic that is of interest to combinatorialists. There will be a dinner afterwards. Date: Fri, 6 Feb 1998 14:42:07 -0500 (EST) From: Shirley Entzminger-Merritt X-Sender: daisymae(at-sign)schauder To: amc(at-sign)math.mit.edu Subject: APPLIED MATHEMATICS COLLOQUIUM -- Monday, February 9, 1998 A REMINDER . . . APPLIED MATHEMATICS COLLOQUIUM TOPIC: SPECTRA OF LAPLACIANS FOR SIMPLICIAL COMPLEXES SPEAKER: PROFESSOR VICTOR REINER School of Mathematics University of Minnesota ABSTRACT: This talk will discuss some recent work on the spectra of discrete Laplace operators coming from boundary maps in a simplicial complex. For two families of simplicial complexes, the chessboard complexes (studied by J. Friedman and P. Hanlon) and matroid complexes (studied by W. Kook, D. Stanton and myself), these spectra are known to be integral, and interpretations of the spectra have been given. Why these particular complexes should have integral spectra is still mysterious, as is the connection to spectra of Laplacians on Riemannian manifolds. One corollary to the interpretation of the spectra for matroid complexes may be paraphrased as stating that one can "hear" the chromatic polynomial of a graph. DATE: MONDAY, FEBRUARY 9, 1998 TIME: 4:15 p.m. LOCATION: Building 2, Room 105 Refreshments will be served at 3:45 p.m. in Building 2, Room 349 Applied Math Colloquium: http://www-math.mit.edu/amc/spring98 Math Department: http://www-math.mit.edu Massachusetts Institute of Technology Department of Mathematics Cambridge, MA 02139 ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From sara(at-sign)math.mit.edu Tue Feb 10 15:41:59 1998 Return-Path: Received: from schubert.mit.edu (SCHUBERT.MIT.EDU [18.87.0.16]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id PAA18344 for ; Tue, 10 Feb 1998 15:41:59 -0500 (EST) Received: (from sara(at-sign)localhost) by schubert.mit.edu (8.8.7/8.8.7) id PAA16854; Tue, 10 Feb 1998 15:41:57 -0500 (EST) Date: Tue, 10 Feb 1998 15:41:57 -0500 (EST) Message-Id: <199802102041.PAA16854(at-sign)schubert.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: pretalk X-URL: http://www-math.mit.edu/~sara Back by popular demand, there will be a pretalk tomorrow from 3:30pm-4:00pm in conjunction with the combinatorics seminar. Alex Postnikov will give a brief introduction to flag manifolds. Recall from last year, we had several pretalks aimed at graduate students and non-experts in the particulars of the seminar that followed. Everyone is welcome to attend. Questions are strongly encouraged! Pretalks will be held in 2-338 (seminar room). If anyone scheduled to give a talk would like to give a pretalk, please let me know. ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From gcrota(at-sign)earthlink.net Tue Feb 10 17:29:15 1998 Return-Path: Received: from germany.it.earthlink.net (germany-c.it.earthlink.net [204.250.46.123]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id RAA22410 for ; Tue, 10 Feb 1998 17:29:12 -0500 (EST) Received: from [153.35.78.167] (1Cust39.max19.boston.ma.ms.uu.net [153.35.78.167]) by germany.it.earthlink.net (8.8.7/8.8.5) with ESMTP id OAA07445 for ; Tue, 10 Feb 1998 14:28:40 -0800 (PST) Date: Tue, 10 Feb 1998 14:28:40 -0800 (PST) Message-Id: Mime-Version: 1.0 Content-Type: multipart/mixed; boundary="============_-1324990680==_============" To: combinatorics(at-sign)math.mit.edu From: Gian-Carlo Rota Subject: colloquium lectures --============_-1324990680==_============ Content-Type: text/plain; charset="us-ascii" Dear colleagues, I am sending you as an attachment the draft of the Colloquium Lectures I delivered at the annual meeting of the American Mathematical Society last January. I will greatly appreciate any comments, corrections, suggestions etc. before I put them on the web. Gian-Carlo Rota --============_-1324990680==_============ Content-Type: text/plain; name="colloquium_2=98"; charset="us-ascii" Content-Disposition: attachment; filename="colloquium_2=98" \documentstyle[12pt]{article} \def\bphiz#1{\overline\phi_{\hat0}^{#1}} \def\bPhi{\overline\Phi} \def\phiz#1{\phi_{\hat0}^{#1}} \def\sigalg#1{\Sigma^{\otimes{#1}}} \def\sigalgpi#1{\Sigma^{\otimes{#1}}_\pi} \def\finalg#1{\Sigma^{\otimes{#1}}_{\rm fin}} \def\finalgpi#1{\Sigma^{\otimes{#1}}_{{\rm fin},\pi}} \def\ltimes{{\rm X}} \def\D{{\cal D}} \def\F{{\cal F}} \def\A{{\cal A}} \def\B{{\cal B}} \def\E{{\cal E}} \def\C{{\cal C}} \def\G{{\cal G}} \def\R{{\bf R}} \def\S{{\bf S}} \def\Inpi{{I^{[n]}_\pi}} \def\bs{\backslash} \def\zpione{{\hat 0 \le \pi \le \hat 1}} \def\zpilone{{\hat 0 \le \pi < \hat 1}} \def\halmos{{Q.E.D.}} % This is temporary \def\St{{\rm St}} \newtheorem{definition}{Definition} \newtheorem{lemma}{Lemma} \newtheorem{theorem}{Theorem} \begin{document} \setlength{\textwidth}{6in} \setlength{\textheight}{8in} \parindent=0pt \tolerance=6000 \title{The American Mathematical Society Colloquium Lectures, 1998} \author{Gian-Carlo Rota} \date{Baltimore, January 7,8 and 9, 1998} \maketitle \newpage \begin{center} {\bf INTRODUCTION TO GEOMETRIC PROBABILITY\\ being\\ The first of three colloquium Lectures\\ delivered at the Annual Meeting of the American Mathematical Society\\ Baltimore, January 7, 1997\\ Gian-Carlo Rota\\ Department of Mathematics\\ MIT\\ Cambridge MA 02139-4307\\ \end{center} \bigskip \bigskip I am very happy to be here before you as the Colloquium Lecturer for this year, and I feel deeply honored to be given this great opportunity to share with you some of the mathematics we love. \bigskip When I received from Bob Fossum the invitation to be the Colloquium Lecturer for this year, I rushed to the library to read in an old issue of the Notices the list of all previous Colloquium Lecturers, going all the way back to James Pierpont in 1896. It is a list of distinguished mathematicians, and I wondered how my name could ever belong in such lofty company. My immediate temptation was to decline the invitation outright; but Bob Fossum assured me that no one in the history of the Society has ever declined the invitation to be the Colloquium Lecturer. So I went back to the list of previous colloquium speakers, in search for a justification of my presence in that list. As often happens in such situations, I soon enough found such a justification. I computed the average age of colloquium lecturers and discovered that this average is somewhat lower than my age, as a matter of fact my age exceeds by approximately by one standard deviation the mean age of previous speakers. As I came to this realization, I began to fantasize on the probable topics that my younger predecessors might have chosen. I imagined a brilliant young mathematician, eager to establish himself as a leader in his field, delivering one single dazzling proof beginning with the first colloquium lecture and lasting all the way to the end of the third. Or else, some middle aged mathematician, anxious to have his latest theory accepted by the mathematical world, delivering to a thrilled audience a three-hour condensation of material that would normally take an entire term in an advanced graduate course. How could I, a mathematician one standard deviation older, ever hope to match such enviable feats? \bigskip These fantasies came to an abrupt end when Bob Fossum informed me in no uncertain terms that the Council of the Society had decided that the three colloquium lectures must deal with three independent and unrelated topics, thereby allowing any member of the audience to skip one or more lectures, without missing anything. Bob Fossum's command deprived me of all possible role models among previous colloquium lecturers. In a state of temporary panic, I again scanned the list of previous colloquium speakers, this time looking for names of mathematicians who had not been chosen for this honor. Sure enough, one name was conspicuously missing: that of Hermann Weyl. \bigskip I hope you will forgive me if I digress with some personal reminiscences. \bigskip In the fall of 1950 I enrolled as a freshman at Princeton, having graduated a few months before from the American High School of Quito, Ecuador. The principal of the American High School of Quito was a Princeton graduate, and he steered me towards Princeton University. \bigskip In November 1950 I listened to my first mathematics lectures. These were the three Vanuxem lectures, delivered by Hermann Weyl and bearing the generic title "Symmetry". They were an unforgettable experience. The lectures took place in the old chemistry auditorium, packed with an expectant public. As I shamelessly sat in the first row trying to guess which of the other persons sitting in the same row was to be the speaker, a hush fell upon the audience: Einstein was entering the lecture room. To my disappointment, he sat somewhere in the middle of the auditorium. \bigskip The first lecture began with an impressive and lengthy quotation in Greek, which no one in the audience understood except Luther Pfahler Eisenhart. This brilliant start was followed by a display of slides portraying charming women wearing the long brimmed hats fashionable at the time, and later by more slides showing the Alhambra and the Pentagon. Not a word of mathematics. The audience was left wondering where such a sparkling display of "Kultur" was leading up to. Not much more mathematics was mentioned in the second lecture, when more slides were shown of physics experiments, for which the lecturer provided a learned oral commentary. Only in the last lecture did some group theory make a modest appearance. By that time the audience, which had not dwindled, was enthralled with the subject, and did not mind the fact that the speaker had said very little about mathematics, actually he had said very little about anything at all. What is more remarkable, the audience seemed to be thankful to the speaker for making the contents of the three lectures independent of one another, thereby minimizing all memory requirements. I hazard to guess that the success of Hermann Weyl's lectures may be in part attributed to the speaker's astute foresight in making his lectures self contained, independent and lightweight. As I recall this distant episode, I realize that Bob Fossum's injunction about the independence of the present colloquium lectures is a wise one, all the more so when the speaker is not Hermann Weyl. \bigskip You may wonder why I cited my age as an aid to delivering these colloquium lectures. What difference does one standard deviation make? I think it makes some difference. It is a relief, both to you and to me, to know right at the start that the speaker does not feel the need to impress you by stating the results of his latest research. Nor do you or I suffer from any lack of exposure to the latest fashions in mathematics; we'll hear enough about them in other lectures scheduled to be delivered at this meeting. We can therefore afford to spend these three hours on leisurely discussion of some mathematics that may matter to both you and me. \bigskip We will cover in these lectures a few items that are not widely known, that should be better known, and that I vouch can be understood by anyone with a B.A. in mathematics. I solemny promise that I will not state any big theorems, I will not subject you to any ingenious arguments, and that I will not announce any revolutionary developments. \bigskip The title of this lecture is "Geometric Probability". A definition of geometric probability might run as follows: geometric probability is the study of invariant measures. Like all definitions, this does not tell us anything until we are shown some typical examples, and these examples are the content of this lecture. \bigskip About one hundred years ago, the properties that underlie such notions as length, area , volume , as well as the probability of events were abstracted under the banner of the word "measure". Let us review the definition of measure, since we will be using this definition in an unusual way. \bigskip A measure $\mu$ is a function defined on a family of subsets of a set $S$, which takes real values, not necessarily positive. The family of sets on which a measure is defined is closed under unions and intersections, and contains the empty set. \bigskip A measure is characterized by two simple axioms. Let us take a minute to review these axioms. \bigskip Axiom 1. $$ \mu ( \emptyset ) = 0,$$ where $\emptyset $ is the empty set. \bigskip Axiom 2. If $A$ and $B$ are two measurable sets, then $$\mu(A\cup B) = \mu (A) +\mu(B) - \mu (A\cap B) .$$ \bigskip The meaning of this second axiom is clear from the picture. The axiom states that measure is additive. In particular, if we have two disjoint sets $A$ and $B$, then $$\mu(A\cup B) = \mu (A) + \mu (B) .$$ \bigskip More generally, for any finite family $F$ whose members are sets, and for which any two members are disjoint, we we have : $$\mu( \bigcup_{A\in F} A ) =\sum_{A\in F}\mu (A).$$ We most emphatically do not assume that a measure is countably additive. \bigskip The best known example of a measure is the volume $\mu_n(A)$ of a solid $A$ in ordinary $n$-dimensional Euclidean space. The volume $\mu_n (A)$ of a solid $A$ satisfies axioms 1 and 2 above, but axioms 1 and 2 do not characterize volume among all possible measures. What additional axioms must we add to the definition of a measure, in order to characterize volume? It is possible to characterize volume among all measures by adding to axioms 1 and 2 two additional intuitive axioms, namely, the following: \bigskip Axiom 3. \bigskip The volume of a set $A$ is independent of the position of $A$. If a set $A$ in $n$-dimensional Euclidean space can be rigidly moved onto a set $B$, then $A$ and $B$ have the same volume. \bigskip In other words, volume is invariant under the group of Euclidean motions. Lastly, we must prescribe a normalization, as physicists say. This is done by taking a parallelotope $P$ with orthogonal sides of lengths $ x_1, x_2, \dots , x_n $, and setting \bigskip Axiom 4. $$ \mu_n (P) = x_1x_2 \dots x_n.$$ \bigskip These axioms, together with suitable continuity conditions, uniquely determine the volume of solids in Euclidean $n$- space. For example, starting from these four axioms, by a limiting process such as one finds in an advanced calculus textbook, one establishes the fact that that the volume of a sphere $S_r$ of radius $r$ in $n$-dimensional space is given by the following formulas: $$\mu_n(S_r) = \frac{ \pi^{n/2} r^{n/2}}{(n/2)!} $$ if the dimension $n$ is even and $$\mu_n (S_r) = \frac{2^n \pi^{(n-1)/2}((n-1)/2)! r^n}{n!} $$ if the dimension $n$ is odd. \bigskip It is still widely believed that volume is the only invariant measure in Euclidean $n$-space. But in point of fact, there are other invariant measures, defined on all reasonable subsets of Euclidean $n$-space, which have a notable geometric significance. Our objective is to describe all such invariant measures. What happens if we keep the first three axioms, but tamper with the fourth axiom, the normalization axiom ? Will we get something interesting, or will we get nothing new? To answer this question, we will appeal to the basic tools of combinatorial mathematics. \bigskip The basic tools of combinatorial mathematics are the elementary symmetric functions, to wit, the following polynomials in $n$ variables: $$e_1(x_1,x_2,...x_n) = x_1 + x_2 + ... +x_n.,$ $$e_2(x_1,x_2,...x_n) = x_1x_2+x_1x_3+...+x_{n-1}x_n,$$ $$ \dots $$ $$e_{n-1}(x_1, x_2, ..., x_n) = x_2x_3...x_n + x_1x_3x_4...x_n+...+ x_1x_2...x_{n-1},$$ $$ e_n(x_1, x_2, ..., x_n) = x_1 x_2 ... x_n .$$ Observe an interesting coincidence. The last of these three symmetric functions is also the formula for the volume of a parallelotope. Axiom 4 can be rewritten as Axiom 4. $$\mu_n (P) = e_n(x_1,x_2,\dots , x_n).$$ \bigskip Let us try an experiment, and replace the $n$-th symmetric function by the $n-$first symmetric function. Let us first take $n = 3$, that is, three-dimensional space, so that we can better visualize what will be going on. Let us see whether we can define a measure on subsets of $3$-dimensional space by keeping three of the above axioms, but by replacing the normalization Axiom 4 by using another symmetric function instead of the symmetric function $e_3(x_1,x_2,x_3)$ which gives the volume. Let us first replace the symmetric function $e_3$ by the symmetric function $e_2$, thereby changing Axiom 4 to \bigskip Axiom 4': $$\mu_2(P) = x_1x_2 + x_1x_3 + x_2x_3 . $$ Does this axiom define a measure ? Of course it does. The right hand side is the formula for the surface area of the parallelotope $P$, divided by $2$. Again we will find in any advanced calculus textbook the explanation of the fact that axioms 1, 2, 3, and 4' , together with some continuity considerations, completely determine an invariant measure which is the surface area of solids in ordinary space. For example, the following well known formula for the surface area of a sphere $S_r$ of radius $r$ in $3$ dimensions is obtained from these axioms: $$\mu_2(S_r)= 4\pi r^2 .$$ Let us take the next step. \bigskip Emboldened by our success with two symmetric functions, we now replace axiom 4 by yet another axiom, using another symmetric function. Let us set \bigskip Axiom 4". $$ \mu_1(P) = e_1(x_1, x_2, x_3) = x_1 + x_2 + x_3 $$ The new measure $\mu_1$ will satisfy axioms 1, 2, and 3, and in addition it sill satisfy axiom 4". The symmetric function of degree one plays the role that in the previous two examples was played by the other two symmetric functions. \bigskip But wait a minute: is this definition consistent? To realize that the definition of the new measure $\mu_1$ is consistent, that is, that $\mu_1$ as defined by axioms 1, 2, 3, and 4" really exists and is not a dream of reason, look at two parallelotopes $P_1$ and $P_2$ that have a face in common. The first parallelotope has sides equal to $x_1, x_2, x_3$ , and the second parallelotope has sides equal to $y, x_2, x_3$. The two parallelotopes have a common face with sides equal to $x_1, x_2$. The measure $\mu_1(P_1\cup P_2)$ of the parallelotope $P_1\cup P_2 $ can be computed in two ways: using the left side of axiom 2, or using the right side, and the two computations had better yield the same answer, in symbols: $$\mu_1(P_1 \cup P_2) = \mu_1(P_1) + \mu_1(P_2) - \mu_1(P_1 \cap P_2)$$ Let us check this. The left side is computed by observing that the parallelotope $P_1\cup P_2$ has sides equal to $x_1$ ,$ x_2$ and $x_3 + y$. Therefore , Axiom 4" tells us that $$\mu_1(P_1 \cup P_2) = x_1 + x_2 + x_3 + y.$ Now let us compute the right side . We have $$\mu_1(P_1) = x_1+x_2+x_3$$ $$\mu_1(P_2) = x_1 + x_2 + y$$ $$\mu_1(P_1 \cap P_2) = x_1 + x_2 ,$$ again by Axiom 4" applied to $P = P_1 \cap P_2$ , since one side equals zero when the parallelotope is a flat, that is, a rectangle. Therefore, the right side of axiom 2 equals $$\mu_1(P_1) + \mu_1(P_2) - \mu_1(P_1 \cap P_2 )= x_1+x_2+x_3 + x_1+x_2+ + y - (x_1+x_2) = $$ $$ x_1+x_2+x_3+y ,$$ and the two sides of our equations agree thereby convincing us that the definition may well be consistent. \bigskip The preceding argument is convincing, even though it proves nothing. Actually, the definition of $\mu_1(P) $ for a parallelotope $P$ has a simple geometric interpretation. When multiplied by 4, it equals the perimeter of the parallelotope $P$, that is, the sum of the lengths of all the edges of the parallelotope $P$. Just as happens for volume and area, it can be shown by continuity considerations that the measure $\mu_1$ can be extended to all reasonable solids in ordinary space, for example, to all convex sets and to all polyhedra, convex or non convex. But, one may object, $\mu_1(P)$ makes sense for a parallelotope $P$, because a parallelotope has a well defined perimeter. What if $A$ is a solid that does not have a well defined perimeter, a sphere for example? The definition of the measure $\mu_1(A) $ for solids $A$ that may not have a well defined perimeter flies in the face of common sense. \bigskip Einstein wrote : "Common sense is the residue of those prejudices that were instilled into us before the age of seventeen". Common sense must constantly readjust to reality. \bigskip The new measure $\mu_1$ that we obtain in this way is called the mean width, a misnomer that has been kept for historical reasons. The mean width of a solid in space is completely characterized by axioms 1, 2, 3, and 4". In particular, it is invariant , that is, it does not depend on position. For example, the formula for the mean width of a sphere of radius $r$ is computed to be $$\mu_1(S_r) = 4r. $$ Thus we see that in three dimensions each of the three elementary symmetric functions of three variables leads to an invariant measure that enjoys equal rights with volume. The first two of these measures are well known, namely, volume and area. The third, the mean width, is at present almost totally unkwnown. I know of no person who has an intuitive feeling for the mean width, similar to the intuitive feeling we have for volume and area. \bigskip Let us conjecture a possible application of the mean width. A potato grower knows that a potato's volume is important, because it determines the nutritional content of the potato. The potato grower also knows that the surface area of a potato is important, because it is rumored that the vitamins in a potato are concentrated in the skin. We may conjecture that as soon as the potato grower will become aware of the mean width, he or she will find a nutritional interpretation of the mean width of a potato. I am indebted to Steve Schanuel for this example. \bigskip A similar kind of reasoning works in $n$ dimension. We discover $n$ different invariant measures, each of them well defined on all polyhedra and on all finite unions of compact convex sets. Each of the $n$ elementary symmetric functions of $n$ variables leads to the definition of a new invariant measure which is a different generalization of the notion of volume. These $n$ measures are called the intrinsic volumes. The intrinsic volumes are first defined on an orthogonal polytope $P$ whose sides equal $x_1,x_2,\dots , x_n$ by setting $$\mu_k(P) = e_k(x_1,x_2,\dots ,x_n),$$ where $ e_x(x_1,x_2,\dots ,x_n) $ is the $k$-th elementary symmetric function. Here, the subscript $k$ ranges from $1$ to $n$. One then proceeds to extend the definition of the intrinsic volumes to more general sets, by a technique which we will shortly see. The intrinsic volumes are independent of each other, except for certain inequalities they satisfy. Mathematicians are presently working on determining these as yet unknown inequalities among the intrinsic volumes. These inequalities generalize the classical isoperimetric inequality that relates volume to area. At present, we know very little about the intrinsic volumes; they have not been around for long and very little research has been done on them. We do not even know the formula for the intrinsic volumes of an $n$-simplex. \bigskip Now you are thinking: this is all fine and dandy, but how is the extension of the intrinsic volumes from parallelotopes to more general sets carried out? And besides, isn't there any intuitive interpretation we can give the intrinsic volumes? \bigskip We will answer both these questions simultaneously. Let us go back to three-dimensional space. You all know that the set of all straight lines in space - not necessarily through the origin - forms a nice algebraic variety, called the Grassmanian. The group of all Euclidean rigid motions acts on the Grassmanian, and there is an invariant measure on the Grassmanian under the action of the group of Euclidean motions. This invariant measure is unique except for a constant factor. A similar statement may be made about the set of all planes, and more generally for the set of all linear varieties of dimension $k$ in Euclidean space of dimension $n$. Remember that these linear varieties need not pass through the origin. \bigskip In the practice of mathematics, computation with invariant measures on Grassmanians is rare ; most mathematicians would be hard put even to recall an explicit formula for the invariant measures on Grassmanians. Let us take a few minutes to get a feeling for the invariant measure on the set of all straight lines in three-space. As is customary, we begin by giving this measure a name: let us call it $\lambda^3_1$; the upper index $3$ stands for three-dimensional space, and the lower index stands for the dimension of a line, namely, one. To repeat, we use the notation $\lambda^3_1$ to denote the invariant measure on the set of all straight lines in three-space. \bigskip Consider a rectangle $R$ placed anywhere in space, and consider the set of all straight lines that meet the rectangle $R$. Can we compute the measure of this set of lines without knowing the formula for the invariant measure on the Grassmanian of all lines in three-space? Of course we can. A straight line meets the rectangle $R$ either at a point or not at all; therefore, the value of the measure of the set of all lines meeting $R$ depends only on the area $\mu_2(R)$ of the rectangle $R$. If we take another rectangle $R'$ whose area is double the area of $R$, then the measure of the set of all lines meeting $R'$ is double the measure of the set of all lines meeting $R$. Proceeding along these lines, we get to Cauchy's functional equation, and we infer that the measure of the set of all straight lines meeting a rectangle $R$ equals a constant times the area $\mu_2(R)$. Since we are at liberty to choose a normalization of the measure, let us agree to set this constant equal to one. \bigskip But instead of working with a rectangle we could have worked with any planar figure $C$ whatsoever, placed in an arbitrary position in space. The measure of the set of lines meeting $C$ equals the area $\mu_2(C)$, by the same reasoning. We stress the assumption that $C$ must lie in a plane. To conclude: even without knowing the formula for the invariant measure $\lambda^3_1$, we can nevertheless compute the value of such a measure on certain sets of lines. \bigskip Let us now take a more sophisticated set of straight lines. We take a set $D$ in three-space which is the union of disjoint sets $C_1, C_2,\dots C_n$, where each of the $C_i$ is contained in a different plane, and we ask for the measure of the set of all straight lines meeting $D$. Such a computation can be carried out, but it is a combinatorial nightmare; so much so, that we are forced to do what mathematicians do when confronted with combinatorial nightmares: they change the problem ever so slightly. In this case we take a hint from the way probabilists work. Let $X_D(\omega)$ equal the number of times the straight line $\omega$ meets the set $D$. Instead of computing a measure, let us compute the integral $$ \int X_D(\omega) d\lambda^3_1(\omega) ,$$ where $\omega$ ranges over the Grassmanian, that is, over the set of all straight lines in space. We will see that we can compute this integral without knowing the measure $\lambda^3_1$ on the Grassmanian. Since $$ D = \bigcup_{i=1}^n C_i ,$$ and since the $C_i$ are disjoint, we have $$ \int X_D(\omega) d\lambda^3_1(\omega) = \sum_{i=1}^n \int X_{C_i}(\omega) d\lambda^3_1(\omega). $$ But we have chosen each of the sets $C_i$ to lie in a plane, so that a straight line meets $C_i$ either once or not at all. It follows that $$ \int X_{C_i}(\omega) d\lambda^3_1(\omega) = \mu_2(C_i) $$ and therefore $$ \int X_D(\omega) d\lambda^3_1(\omega) = \sum_{i=1}^n \mu_2(C_i) . $$ What is this identity telling us? The right hand side equals the area of the surface $D$. Nothing stops us from passing to the limit, and making the following assertion. Let $E$ be "any" surface in space, and let $X_E(\omega) $ be the number of times the straight line $\omega$ meets the surface $E$. Then the integral $$ \int X_E(\omega) d\lambda^3_1(\omega) $$ ranging over all straight lines $\omega$, equals the surface area of $E$. In probabilistic language: the average number of times a randomly chosen straight line meets the surface $E$ equals the surface area of $E$. \bigskip Let us now retrace our steps, and repeat the same reasoning taking the set of all planes in space, instead of the set of all straight lines. The invariant measure on this Grassmanian is denoted by $\lambda_2^3$, where again the upper index stands for three dimensional space, and the lower index for the dimension of a plane. Since a plane meets a straight line segment either at a point or not at all, the same argument shows that the measure of the set of all planes that meet a line segment $L$ equals $\mu_1(L)$, namely, the length of the segment $L$; more generally, if $F$ is any curve "whatsoever" in space, and if $X_F(\omega)$ equals the number of times the plane $\omega$ meets the curve $F$, then repeating the argument we used for straight lines we infer that the integral $$ \int X_F(\omega) d\lambda^3_1(\omega) $$ equals the length of the curve $F$. The variable of integration $\omega$ now ranges over planes, not over straight lines. Here again we compute an integral without knowing the measure. \bigskip We are now very close to getting an intuitive interpretation of the mean width. Recall the parallelotope $P$ with sides equal to $x_1,x_2,x_3$. Let us take the curve to be the perimeter of the parallelotope $P$. Every plane meets the perimeter of the parallelotope at four points. Therefore, the measure of the set of all planes that meet the perimeter of the parallelotope equals four times the perimeter of the parallelotope. But a plane meets the perimeter of a parallelotope if and only if it meets the parallelotope. We therefore reach an important conclusion: the measure of the set of all planes meeting a parallelotope equals the mean width of the parallelotope, except for a constant factor which we will again set to be one. \bigskip In view of this realization, we can immediately see how to define the mean width of any closed convex set: it equals the measure of the set of all planes that meet the convex set. Thus, we have shown that the mean width may be extended to all closed convex sets in space. \bigskip We are now in a position to give a probabilistic interpretation of the mean width of a convex set. Take two compact convex sets $A$ and $B$ in three dimensional Euclidean space, and suppose that $A$ is contained in $B$. Let us begin by belaboring the obvious. Suppose that we take a point at random belonging to the larger set $B$. What is the probability that the point shall belong to the smaller set $A$? The answer is clear: such a probability equals the ratio of the volume of $A$ by the volume of $B$. \bigskip Instead of choosing a point at random, let us choose a straight line at random in space. Assuming that such a straight line meets the larger set $A$, what is the probability that such a straight line will also meet the smaller set $B$? We have already computed the answer to this question, albeit implicitly. Such a probability equals the surface area of the set $A$, divided by the surface area of the set $B$. \bigskip You can tell what is coming next. We now take a random plane in space. Assuming that the plane meets the larger set $B$, what is the probability that it will also meet the smaller set $A$? The answer is the following: such a probability equals the mean width of $A$, divided by the mean width of $B$. \bigskip In Euclidean $n$-space, we obtain by much the same reasoning interpretations of the intrinsic volume $\mu_k(C)$ of a compact convex set $C$ as the Grassmanian measure of the set of all linear varieties of dimention $n-k$ that meet the convex set $C$, and a similar probabilistic interpretation holds. \bigskip What comes next? There are at least two questions still open. First, are there any other invariant measures besides the intrinsic volumes, and second, how can the definition of the intrinsic volumes be extended to more general subsets of $n$-space than convex sets. The answers to both these questions are closely related. \bigskip The answer to the first question is negative. We are missing one measure, and to discover it, we will engage for a minute in the kind of mathematical reasoning that physicists find unbearably pedantic, just to show physicists that such reasoning does pay off. Let us ask ourselves the question: what is the value of the symmetric function of order zero of a set of $n$ variables $x_1,x_2,\dots , x_n$, or $e_0(x_1,x_2,\dots , x_n)$? I will give you the answer, and will leave it to you to justify this answer after the lecture is over. The answer is the following: $e_0= 1$ if $n >0$, that is, if the set of variables $x_1,x_2,\dots , x_n$ is non empty, and $e_0 = 0 $ if the set of variables is empty. \bigskip We are led to believe that there may exist an invariant measure in $n$-space associated with the symmetric function of order zero. We set $$\mu_0(C) = 1 $$ if $C$ is any non empty compact convex set, and of course $\mu_0(\oslash ) = 0$. Does such a measure exist? It does indeed exist, and the fact that it exists is, in my opinion, one of the most remarkable discoveries ever made in mathematics. \bigskip We will prove that such a measure is well defined on any set which is a finite union of compact convex sets. We do this by employing a classical device borrowed from functional analysis: instead of defining a measure, we define a linear functional on all simple functions, that is, on all real functions $f(\omega)$ defined for $\omega \in R^n $ which are linear combinations of indicator functions of compact convex sets. Let us first begin with the case $n=1$, that is, let $\omega$ range over points on the line. Define a linear functional $\chi_1$ on simple functions as follows: $$ \chi_1(f) = \sum (f(\omega) - f(\omega+)) ,$$ where the sum ranges over all real numbers $\omega$. The meaning of the plus sign is best gleaned from an example. Let f be the indicator function of the closed segment $[a,b]$. Then $f(\omega) - f(\omega+) = 0$ for all $\omega $ except $\omega = b $, because we have $f(b) = 1$ but $f(b+) = 0$. Thus, we see that $\chi_1(f) = 1 $ if $f$ is the indicator function of an interval $[a,b]$. \bigskip Now let us go over to $n$ dimensions, proceeding by induction. Do not worry, this won't take long. Take a straight line $L$ and for every point $\omega$ in $L$ let $H_{\omega} $ be the hyperplane through the point $\omega $ perpendicular to the line $L $. If $f$ is a simple function defined in $n$ space, and if $\omega$ is a point on the straight line $L$, let $f_{\omega}$ be the restriction of $f$ to the hyperplane $H_{\omega} $. Define a linear functional $\chi_n $ as follows: $$ \chi_n(f) = \sum \chi_{n-1}(f_{\omega}) - \chi_{n-1}(f_{\omega +}) ,$$ where the sum ranges over all points $\omega$ on the line $L$. There is only a finite set of $\omega's $ for which the summand is non zero. When $f$ is the indicator function of a non empty compact convex set, then an argument similar to the preceding shows that $\chi_n(f) = 1 $. Thus, we may define a measure $\mu_0(G) = \chi_n(f) $, where $G$ is any finite union of compact convex sets, and $f$ is the indicator function of the set $G$. We have thus proved the existence of a measure $\mu_0$ which is defined on all finite unions of compact convex sets, and which takes the value one on all non empty compact convex sets . This measure has a long history: it is the Euler characteristic. \bigskip Now you are thinking: if this is the Euler characteristic, then it is up to you to show that it coincides with what we ordinarily believe to be the Euler characteristic. Let us conclude this lecture by deriving the formula of Euler-Schl\"afli-Poincar\'e formula for polyhedra. As a matter of fact, this formula can be encapsulated into a simpler formula, one that is easy to remember. Let $C$ be a non empty compact convex polytope of dimension $n$, and let $int(C)$ be the interior of $C$. Then we have the following fundamental formula for the Euler characteristic of $int(C)$: $$ \mu_0(int(C)) = (-1)^n .$$ Indeed, if $f$ is the indicator function of the set $int(C)$, we have: $$ \mu_0(int(C)) = \sum \chi_{n-1}(f_{\omega}) - \chi_{n-1}(f_{\omega + }) ,$$ where the sum ranges over all points $\omega$ on the line $L$ as above. But by induction, we see that every term on the right hand side equals zero, except when $\omega$ is the first point on the line $L$ for which the intersection $C \cap H_{\omega} $ is not empty . If $\omega_{\ell} $ is such a first point, then we have $$ \chi_{n-1}(f_{\omega_{\ell}}) = 0 $$ because the point $\omega_{ell} $ is on the boundary of $C$, and $$ \chi_{n-1}(f_{\omega +}) = (-1)^{n-1} $$ by induction hypothesis,because $ f_{\omega +}$ is the indicator function of the set $int(C) \cap H_{\omega} $, which is the interior of a convex polyhedron one dimension lower. Putting all this together, we obtain $$ \mu_0(int(C)) = \sum \chi_{n-1}(f_{\omega_{\ell}}) - \chi_{n-1}(f_{\omega_{\ell}+}) = - (-1)^{n-1} = (-1)^n ,$$ as desired. We are now in a position to state the famous Euler formula for polyhedra. What is a polyhedron? A polyhedron is a finite union of convex polyhedra. Given a polyhedron, we must define a system of faces. We will say that a set $\bf{F}$ of convex polyhedra is a system of faces for an arbitrary polyhedron $K$ when the elements of $\bf{F}$, called faces, are non empty compact convex sets $F$ with disjoint interiors such that $$ K = \bigcup_{F \in \bf{F} }int(F). $$ Caution: the interior of a face of dimension $k$ is to be taken relative to the linear space of dimension $k$ that contains the face, and the interior of a point is a point. Under these conditions we may take the Euler characteristic of both sides, and using the fact that any two interiors of faces are disjoint we obtain (using the fact tghat the measure of the disjoint union of a family of sets equals the sum of the mesaures of the individual sets): $$ \mu_0(K) = \sum_{F \in \bf{F} } \mu_0(int(F)) = f_0 - f_1 + f_2 - \dots + \dots , $$ where $f_i$ equals the number of faces of dimension $i$. This is Euler's formula. \bigskip We can now answer the second of the questions we had left open: how to extend the definition of the intrinsic volumes from compact convex sets to all finite unions of compact convex sets. If $G$ is such a finite union of compact convex sets, then we set $$ \mu_k (G) = \int \mu_0(G \cap \omega ) d\lambda_{n-k}^n(\omega), $$ where $\omega$ ranges over all linear varieties of dimension $n-k$ in $n$-space. The left hand side defines is a measure, and when $G$ is a compact convex sets it agrees with the definition we have already given. It is therefore the desired extension. The Euler characteristic does all the work for us. \bigskip We are now in a position to state the main theorem of geometric probability. We will say that an invariant measure $\mu$ on Euclidean $n$-space, defined on all finite unions of compact convex sets, is continuous, when $$ \lim_{C_n \rightarrow C} \mu(C_n) = \mu(C) $$ For all sequences $C_n$ of compact convex sets converging to the compact convex set $C$. We have the \bigskip {\bf Main Theorem of Geometric Probability} \bigskip The $n+1$ intrinsic volumes $\mu_0, \mu_1,\dots , \mu_n $ are a basis of the space of all continuous invariant measures defined on all finite unions of compact convex sets. \bigskip The first proof of this theorem is due to Hadwiger; the first elementary proof was published last year by Dan Klain of Georgia Tech. \bigskip In closing, let me try to answer the question you are about to ask: what has this got to do with geometric probability, anyway? I will attempt a sketchy answer. Consider two compact convex sets $A$ and $B$. We imagine $B$ to be fixed in $n$-space, and that we "drop" the rigid set $A$ at random. What is the probability that $A$ meets $B$? We answer this question in three steps. First, we realize that by keeping $B$ fixed and varying $A$ by the group of Euclidean Motions, we define an invariant measure on convex sets $B$. Second, we apply Hadwiger's theorem, and infer that such an invariant measure equals a linear combination of the $n+1$ intrinsic volumes, with coefficients independent of $B$. Third, we determine these coefficients by taking suitable $B$'s. The end result is an identity which is known as the kinematic formula, which has been the object of much research in this century, still going on today. \bigskip Thank you for your attention. \newpage \begin{center} {\bf INVARIANT THEORY, OLD AND NEW\\ being\\ The second Colloqium Lecture\\ \bigskip delivered at the Annual Meeting of the American Mathematical Society\\ Baltimore, January 8, 1997.\\ \bigskip Gian-Carlo Rota\\ Department of Mathematics\\ MIT, room 2-351\\ 77 Massachusetts Avenue\\ Cambridge MA 02139-4307\\ \end{center} \bigskip \bigskip Invariant theory is the great Romantic story of mathematics. For one hundred and fifty years, from its beginnings with Boole to the time, around the middle of this century, when it branched off into several independent disciplines, mathematicians of all countries were brought together by their common faith in invariants: in England, Cayley, MacMahon, Sylvester and Salmon, and later, Alfred Young, Aitken, Littlewood and Turnbull. In Germany, Clebsch, Gordan, Grassmann, Sophus Lie, Study; in France, Hermite, Jordan and Laguerre; in Italy, Capelli, Brioschi, Trudi and Corrado Segre, in America, Glenn, Dickson, Carus (of the Carus Monographs), Eric Temple Bell and later Hermann Weyl. Seldom in history has an international community of scholars felt so united by a common scientific ideal for so long a stretch of time. In our century, Lie theory and algebraic geometry, differential algebra and algebraic combinatorics are offsprings of invariant theory. No other mathematical theory, with the exception of the theory of functions of a complex variable, has had as deep and lasting an influence on the development of mathematics. \bigskip Eventually, invariant theory was to become a victim of its own success: the very term "invariant theory" is nowadays understood in such a wide variety of senses that it has become all but meaningless. It is no wonder that you are baffled by the title of this lecture, and curious to hear what will be said about invariant theory in the next forty-eight minutes. \bigskip Like the Arabian phoenix arising from its ashes, classical invariant theory, once pronounced dead, is once again at the forefront of mathematics. The old treatises are being dusted off the shelves of library basements and reread, reinterpreted and presented in a language that meets the standard of rigor of our day. The program of classical invariant theory, that had for some time been given up as hopeless, is again being pursued, and success may at last be within reach. \bigskip We will review two turning points in the history of invariant theory. The first, the "new" one, happened around the turn of the century, and its effects are still being felt all over mathematics. The second, the "old" one, happened very early in the game, and led to a serious misunderstanding that lasts to this day. \bigskip A pedestrian definition of invariant theory might go as follows: invariant theory is the study of orbits of group actions. Such a definition is correct, but it must be supplemented by a programmatic statement. Hermann Weyl, in the introduction to his book "The Classical Groups", was the first in this century to give a sweeping overview of the program of invariant theory. He summarized this program in two basic assertions. The first states that "All geometric facts are expressed by the vanishing of invariants" , and the second states that "all invariants are invariants of tensors". \bigskip Let us briefly comment on these lofty statements. What is a geometric fact? A geometric fact is a fact about space that is independent of the choice of a coordinate system. Geometric facts are described by means of equations which require a choice of coordinates. In a vector space $V$ of dimension $n$ one chooses a coordinate system $ x_1, x_2,\dots ,x_n $ . Since Descartes, we have learned to express geometric facts by equations in the coordinates $ x_1, x_2,\dots ,x_n $. However, about one hundred years ago, mathematicians and physicists made the shocking discovery that the usual type of equations, that is, equations in the commutative ring generated by the variables $ x_1, x_2,\dots ,x_n $, are inadequate for the description of a lot of geometric and physical facts. Motivated by this discovery, they introduced a more general ring. This is the ring of non commutative polynomials in the coordinates $ x_1, x_2,\dots ,x_n $. Homogeneous elements of this ring, that is, homogeneous non commutative polynomials in the variables $ x_1, x_2,\dots ,x_n $, are called tensors. If we believe Hermann Weyl's philosophy, then we will be satisfied that equations in the tensor algebra suffice for the description of any geometric fact we will ever meet. Furthermore, if these equations are to express geometric properties, then they must hold no matter what coordinate system is chosen; in other words, equations that describe geometric facts must be invariant under changes of coordinates. The program of invariant theory, from Boole to our day, is precisely the translation of geometric facts into invariant algebraic equations expressed in terms of tensors. This program of translation of geometry into algebra was to be carried out in two steps. The first step consisted in decomposing tensor algebra into irreducible components under changes of coordinates. The second step consisted in devising an efficient notation for the expression of invariants for each irreducible component. The first step was successfully carried out in this century; the second was abandoned sometime in the twenties and only recently has it resurfaced. \bigskip The decomposition of tensor algebra into irreducible components was discovered around the turn of the century almost simultaneously by Issai Schur and Alfred Young. The gist of this decomposition is one of the great advances in mathematics of all times, and it may be worthwhile to present it in a form that can be made available to undergraduates. \bigskip Let us consider functions of three variables, such as $f(x_1, x_2, x_3)$. Two well known classes of functions of three variables are symmetric functions, defined to satisfy the equations $$f_s(x_1,x_2,x_3) = f_s(x_{i_1},x_{i_2},x_{i_3}) $$ for every permutation sending the indices $(1,2,3)$ to $(i_1,i_2,i_3)$ , and skew symmetric functions , defined by the equations $$f_a(x_1,x_2,x_3) = \pm f_a(x_{i_1},x_{i_2},x_{i_3}), $$ where the sign is $+1$ or $-1$ according as the permutation sending the indices $(1,2,3)$ to $(i_1,i_2,i_3)$ is even or odd. \bigskip It is not true that a function of three variables is the sum of a symmetric function and a skew-symmetric function. A third type of function is required, which is called a cyclic function, which is defined by the equation $$f_c(x_1,x_2,x_3) + f_c(x_3,x_1,x_2) + f_c(x_2,x_3,x_1) = 0.$$ Every function of three variables can be uniquely written as the sum of a symmetric function, a skew symmetric function, and a cyclic function, in symbols $$f(x_1,x_2,x_3) = f_s(x_1,x_2,x_3) +f_a(x_1,x_2,x_3) +f_c(x_1,x_2,x_3).$$ Each of the three symmetry classes is invariant under permutations; this fact is obvious for symmetric and skew symmetric functions but not quite so obvious for cyclic functions. These three invariant subspaces play for the group of permutations of a set of three elements a role analogous to the role of the eigenvectors of a symmetric matrix. \bigskip For functions $f(x_1,x_2,x_3,x_4) $ of four variables there are five symmetry classes, which are defined as follows: \bigskip 1. Symmetric functions. \bigskip 2. Skew symmetric functions. \bigskip 3. Cyclic symmetric functions, satisfying the three equations $$f(x_1,x_2,x_3,x_4)+ f(x_1,x_4,x_2,x_3)+f(x_1,x_3,x_4,x_3) = 0,$$ $$f(x_1,x_2,x_3,x_4)+f(x_4,x_2,x_1,x_3)+f(x_3,x_2,x_4,x_1) = 0,$$ $$f(x_1,x_2,x_3,x_4)+f(x_4,x_1,x_3,x_2)+f(x_2,x_4,x_3,x_1) = 0,$$ $$f(x_1,x_2,x_3,x_4)+f(x_3,x_1,x_2,x_4)+f(x_2,x_3,x_1,x_4)=0.$$ \bigskip 4. Functions satisfying the four equations $$f(x_1,x_2,x_3,x_4)+f(x_2,x_1,x_3,x_4)+f(x_1,x_2,x_4,x_3)+ f(x_2,x_1,x_4,x_3) = 0,$$ $$f(x_1,x_2,x_3,x_4)+f(x_3,x_2,x_1,x_4)+f(x_1,x_4,x_3,x_2)+ f(x_3,x_4,x_1,x_2)= 0,$$ $$f(x_1,x_2,x_3,x_4)+f(x_1,x_3,x_2,x_4)+f(x_4,x_2,x_3,x_1)+ f(x_4,x_3,x_2,x_1)= 0,$$ $$\sum sign(\sigma ) f(x_{\sigma 1},x_{\sigma 2},x_{\sigma 3}, x_{\sigma 4})= 0.$$ \bigskip 5. Functions satisfying the equations $$f(x_1,x_2,x_3,x_4)-f(x_2,x_1,x_3,x_4)-f(x_1,x_2,x_4,x_3)+ f(x_2,x_1,x_4,x_3) = 0,$$ $$f(x_1,x_2,x_3,x_4)-f(x_3,x_2,x_1,x_4)-f(x_1,x_4,x_3,x_2)+ f(x_3,x_4,x_1,x_2)= 0,$$ $$f(x_1,x_2,x_3,x_4)-f(x_1,x_3,x_2,x_4)-f(x_4,x_2,x_3,x_1)+ f(x_4,x_3,x_2,x_1)= 0,$$ $$\sum f(x_{\sigma 1},x_{\sigma 2},x_{\sigma 3}, x_{\sigma 4})= 0.$$ \bigskip Every function of four variables is uniquely expressible as the sum of five functions, each one belonging to one of these symmetry classes. Each symmetry class is invariant under permutations. \bigskip More generally, every function of $n$ variables $f(x_1,x_2,\dots ,x_n)$ can be uniquely written as the sum of $p_n$ functions, each one belonging to a different symmetry class. Here, $p_n$ equals the number of partitions of the integer $n$. Each symmetry class is defined by equations which are not difficult to find. This decomposition holds for tensors as well, after some cosmetic changes of notation. To this day, only two symmetry classes of tensors have been studied in any detail. Symmetric tensors are ordinary commutative polynomials such as we learned to use in analytic geometry. Skew symmetric tensors are polynomials in the coordinates $ x_1, x_2,\dots ,x_n $ when the variables are assumed to satisfy the equations $x_ix_j = - x_jx_i $. Tensors belonging to symmetry classes other than the classes of symmetric and skew symmetric tensors also occur in geometry and physics. However, these symmetry classes have been studied very little, and they are a long way from being understood. \bigskip So much for the word "new" in the title of this lecture; let us next do some justice to the word "old". We will describe the most peculiar feature of classical invariant theory, namely, the symbolic or umbral notation, to which Eric Temple Bell dedicated his Colloquium Lectures in 1927. We will consider the simplest group, namely, the group of translations of the line. The unusual features of the symbolic method will already be apparent in this special case. Let $p(x)$ and $ q(x) $ be monic polynomials in the variable $x$. We write them in the following quaint notation: $$ p(x) = x^n + {n \choose 1}a_1x^{n-1} + {n \choose 2}a_2 x^{n-2} + \dots + {n \choose n-1}a_{n-1}x + a_n , $$ and $$ q(x) = x^k + {k \choose 1}b_1x^{k-1} + {k \choose 2}b_2 x^{k-2} + \dots + {k \choose k-1}b_{k-1}x + b_k . $$ We assume that the polynomial $q(x)$ is of lower degree than the polynomial $p(x)$, that is, that $k \leq n $. Define the translation operator $T^c$ on a polynomial $p(x)$ as follows: $$ T^c p(x) = p(x + c) .$$ Let us write $$ p(x+c) = x^n +{n \choose 1}p_1(c) x^{n-1} + {n \choose 2}p_2(c) x^{n-2} + \dots + {n \choose n-1}p_{n-1}(c) x + p_n(c) , $$ The $j$-th coefficient $p_j(c)$ of the polynomial $p(x+c) $ is computed to be $$ p_j(c) = a_j + {j \choose 1} a_{j-1} c + {j \choose 2 }a_{j-2}c^2 + \dots + c^n . $$ A polynomial I($a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k) $ in the variables $a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k $ is said to be an invariant of the two polynomials $p(x)$, $q(x)$ when $$ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k) = I(p_1(c),p_2(c), \dots , p_n(c), q_1(c), q_2(c), \dots , q_k(c) ) $$ for all complex numbers $c$. By abuse of notation, we write $I(p(x), q(x)) $ and we speak of $I$ as being an invariant of the polynomials $p(x)$ and $q(x)$. In this abusive notation, a polynomial $I$ is said to be an invariant of the polynomials $p(x)$ and $q(x)$ whenever $$ I(T^cp(x), T^cq(x) ) = I(p(x), q(x)) $$ for all constants $c$. \bigskip Invariant theory is concerned with the problem of finding all invariants of a given set of polynomials, as well as their significance. \bigskip What is meant by the "significance" of an invariant? We will appeal to Hermann Weyl. "Every" property of polynomials which is invariant under the group of translations is expressed by the vanishing of a set of invariants. In other words, "any" set of polynomials which is invariant under translations is the same set as a set of polynomials obtained by setting to zero a set of invariants of such polynomials. It is impossible to understand the above statement without examples. Let us consider the simplest and oldest example. The property of a quadratic polynomial $$ q(x) = x^2 + 2b_1 x + b_2 .$$ of having a double root is invariant under translations; in other words, if the polynomial $q(x)$ has a double root, so does the polynomial $p(x+c)$ for any constant $c$ . Following Hermann Weyl, we look for an invariant whose vanishing expresses this property. Sure enough, it is easy to check that the discriminant $$ D( b_1, b_2) = b_1^2 - b_2 $$ is the desired invariant. This example, due to Boole, was the spark that led to the birth of invariant theory. \bigskip One often hears the sentence "Hilbert killed invariant theory", repeated as an excuse to ignore all that went on in invariant theory after Hilbert. I don't know who made up this infamous sentence. It is not true. Hilbert loved invariant theory, and he went on publishing striking papers in invariant theory well after he proved the theorem that is nowadays called the Hilbert basis theorem, the theorem that is supposed to have killed invariant theory. Some of the most fascinating results in invariant theory were discovered in the first twenty years of this century, a long time after Hilbert proved his basis theorem. \bigskip What then is the reason for the temporary demise of invariant theory in this century? One reason is the endemic use of a notation that lacked rigor and that amounted to little more than handwaving in print. This is the symbolic or umbral notation. \bigskip Dieudonn\'e wrote that half the success of a piece of mathematics depends on a proper choice of notation. It would be interesting to make a list of unfortunate notations that killed various chapters of mathematics, as well as a list of felicitous notations that promoted the development of other branches of mathematics. The symbolic or umbral notation that was used by invariant theorists through the nineteen twenties was a catastrophe. A number of mathematicians tried to make sense of the symbolic method without success, the three most notable ones being Hermann Weyl, Eric Temple Bell, and Edward Hegeler Carus. Eric Temple Bell failed to properly define umbral notation, and his book "Algebraic Arithmetic" remains to this day the book of seven seals. If Hermann Weyl and Eric Temple Bell had lived fifty years longer, so as to benefit of the development of what was in their time called "modern" algebra, they would undoubtedly have succeeded in properly defining umbral notation. In our day, it does not take much work to accomplish this task. Do not be alarmed: it will only take a few minutes. Before I start spouting out definitions, let me say what I am not going to say. Umbral notation can be shown to be equivalent, or "cryptomorphic", to use a term invented by my late friend Garrett Birkhoff, to another notation that has gained great notoriety in our day: I mean the notation of Hopf algebras. I will not justify this Sybilline pronouncement, not because it is difficult to do so, but because it would be too boring to do so. \bigskip Let us go on to the definition of umbral notation. Side by side with the polynomials $p(x)$ and $q(x)$, we consider another polynomial algebra ${\bf C}[ x, \alpha , \beta ] $ in three variables $x$ , $ \alpha $ and $\beta $, together with a linear functional $E$ defined on the underying vector space ${\bf C}[x, \alpha, \beta ]$. The definition of the linear functional $E$ is the key point. It is carried out in the following steps: \bigskip Step 1. Set $$ E(x^j ) = x^j $$ for all non negative integers $j$ , in particular $E(1) = 1$. \bigskip Step 2. Set $$ E(\alpha^j) = a_j ,$$ in particular, we have $ E(\alpha^j) = 0 $ if $j > n $. \bigskip Step 3. Set $$ E(\beta^j) = b_j ,$$ in particular, we have $ E(\beta^j) = 0 $ if $j > k $. \bigskip Step 4. This is the main step. Set $$ E(\alpha^i \beta^j x^{\ell}) = E(\alpha^i ) E( \beta^j )x^{\ell}. $$ Following Sylvester, the variables $\alpha $ and $\beta $ are called umbrae. In other words, the linear functional $E$ is multiplicative on distinct umbrae. \bigskip Step 5. Extend by linearity. \bigskip This completes the definition of the linear functional $E$. We next come to the most disquieting feature of umbral notation. Let $f(\alpha , \beta , x) $ and $g(\alpha , \beta , x) $ be two polynomials in the variables $ \alpha , \beta , x $. We write $$ f(\alpha , \beta , x) \cong (\alpha , \beta , x) $$ to mean $$ E(f(\alpha , \beta , x)) = E( g(\alpha , \beta , x) ) . $$ Read $\cong $ as " equivalent to". The "classics" went a bit too far, they wrote $$ f(\alpha , \beta , x) = g(\alpha , \beta , x) $$ that is, they replaced the symbol $\cong$ by ordinary equality. This was an excessive abuse of notation. The "classics" were aware of the error, and while they avoided computational errors by clever artistry, they were unable to settle on a correct notation. \bigskip The umbral or symbolic method consists in replacing all occurrences of the coefficients of the polynomials $p(x)$ and $q(x)$ by umbrae and equivalences. For example, $$ p(x) \cong (x + \alpha)^n $$ and $$ q(x) \cong (x+ \beta)^k .$$ Let us carefully check the first equivalence. By definition, the equivalence means the same as $$ E(p(x)) = E( (x + \alpha)^n).$$ Since $E(x^j) = x^j $ for all non negative integers $j$, this identity can be rewritten as $$ p(x) = E( (x + \alpha)^n).$$ Expanding the right hand side by the binomial theorem, we obtain $$ E( (x + \alpha)^n) = $$ $$ E( x^n + {n \choose 1}\alpha x^{n-1} + {n \choose 2}\alpha^2 x^{n-2} + \dots + {n \choose n-1}\alpha^{n-1}x +\alpha^n) $$ By linearity this equals $$ x^n + {n \choose 1}E(\alpha) x^{n-1} + {n \choose 2}E(\alpha^2) x^{n-2} + \dots + {n \choose n-1}E(\alpha^{n-1})x + E(\alpha^n) . $$ Evaluating the linear functional $E$, we see that this in turn equals $$ x^n + {n \choose 1}a_1x^{n-1} + {n \choose 2}a_2 x^{n-2} + \dots + {n \choose n-1}a_{n-1}x + a_n , $$ as desired. The expression $$ (x + \alpha)^n $$ is called an umbral representation of the polynomial $p(x)$. In umbral notation, a complex number $r$ is a root of the polynomial equation $p(x) = 0$ if and only if $$ (\alpha + r)^n \cong 0. $$ Similarly, in umbral notation the polynomial $T^cp(x)= p(x+c)$ may be represented as follows: $$ p(x+c ) \cong (x+\alpha + c)^n, $$ and this yields the umbral expression for the coefficients $p_j(c)$ of the polynomial $p(x+c)$, namely $$ p_j(c) \cong (\alpha + c)^j . $$ Let us next see how umbral notation is related to invariants. Let us assume that the two polynomials $p(x)$ and $q(x)$ have the same degree $n$. Then an invariant $A$ of the polynomials $p(x), q(x)$ may be defined as follows: $$ A(q(x), p(x)) \cong (\beta - \alpha)^n .$$ The evaluation of the invariant $A$ in terms of the coefficients of $p(x)$ and $q(x)$ proceeds as follows: $$ A(q(x), p(x)) = E(\beta - \alpha)^n) = $$ $$E(\beta^n -{n \choose 1} \beta^{n-1}\alpha + \dots + (-1)^{n-1}{n \choose n-1}\beta \alpha^{n-1} + (-1)^n \alpha_n) = $$ $$ E(\beta^n) - E({n \choose 1} \beta^{n-1}\alpha) + \dots + (-1)^{n-1}E({n \choose n-1}\beta \alpha^{n-1}) + (-1)^n E(\alpha^n)= $$ $$ E(\beta^n) - {n \choose 1} E(\beta^{n-1})E(\alpha) + \dots + (-1)^{n-1}{n \choose n-1}E(\beta) E(\alpha^{n-1}) + (-1)^n E(\alpha^n) = $$ $$ b_n - {n \choose 1} b_{n-1}a_1 + {n \choose 2}b_{n-2}a_2 - \dots + \dots + (-1)^{n-1}{n \choose n-1}b_1a_{n-1} + (-1)^n a_n. $$ Why is $A$ an invariant? This is best seen in umbral notation: $$ A(T^cq(x), T^cp(x)) \cong (\beta + c - \alpha - c )^n = (\beta - \alpha)^n. $$ The invariant $A$ is called the apolar invariant; two polynomials $p(x)$ and $q(x)$ having the property that $A(q(x), p(x)) = 0 $ are said to be apolar. In umbral notation, two polynomials are apolar whenever $$ (\beta - \alpha)^n \cong 0. $$ The concept of apolarity has a distinguished pedigree going all the way back to Apollonius. \bigskip What is the "significance" of the apolar invariant? What does it mean for two polynomials to be apolar? This question is answered by \bigskip {\bf Theorem 1}. Suppose that $r$ is a root of the polynomial $q(x)$, that is, that $q(r)= 0$. Then the polynomials $q(x)$ and $p(x) = (x - r)^n $ are apolar. \bigskip Proof. For $p(x) = (x-r)^n $ we have $\alpha^j \cong (-r)^j $, and hence $$ A(q(x),p(x)) \cong (\beta - (-r))^n = (\beta + r)^n \cong 0, $$ as desired. \bigskip Corollary. If the polynomial $q(x)$ has $n$ distinct roots $r_1,r_2,\dots , r_n $, and if the polynomial $p(x)$ is apolar to $q(x)$, then there exist constants $c_1,c_2,\dots ,c_n $ for which $$ p(x) = c_1 (x-r_1)^n + c_2 (x-r_2)^n + \dots + c_n (x-r_n)^n . $$ \bigskip Proof. The dimension of the affine subspace of all monic polynomials $p(x)$ which are apolar to $q(x)$ equals $n$. But if the polynomial $q(x)$ has simple roots, then by the above theorem the polynomials $(x-r_1)^n, (x-r_2)^n,\dots , (x- r_n)^n $ are linearly independent and apolar to $q(x)$. Hence the polynomial $p(x)$ is a linear combination of these polynomials. This completes the proof. \bigskip Thus, we see that apolarity gives a trivial answer to the following question: when can a polynomial $p(x)$ be written as a linear combination of polynomials of the form $(x- r_1)^n, (x-r_2)^n,\dots , (x-r_n)^n $? \bigskip A beautiful theorem on apolarity was proved by the British mathematician John Hilton Grace. We state it without proof: \bigskip {\bf Grace's Theorem}. If two polynomials $p(x)$ and $q(x)$ of degree $n$ are apolar, then every disk in the complex plane containing every zero of $p(x)$ also contains at least one zero of $q(x)$. \bigskip Grace's Theorem is an instance of what might be called a sturdy theorem. For almost one hundred years it has resisted all attempts at generalization. Almost all known results about the distribution of zeros of polynomials in the complex plane are corollaries of Grace's theorem. \bigskip We will next generalize the apolar invariant to the case of two polynomials $p(x)$ and $q(x)$ of different degrees $n$ and $k$, with $ k \leq n$. To this end, we slightly generalize the definition of invariant, as follows. \bigskip A polynomial $ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k, x ) $ in the variables $a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k , x $ is said to be an invariant of the polynomials $p(x)$, $q(x)$ when $$ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k, x) = $$ $$I(p_1(c),p_2(c), \dots , p_n(c), q_1(c), q_2(c), \dots , q_k(c), x+ c ) $$ for all complex numbers $c$. \bigskip Sometimes these more general invariants are called covariants. \bigskip We define a more general apolar invariant as follows: $$ A(q(x), p(x)) \cong (\beta - \alpha)^k (x- \alpha )^{n-k}. $$ Again, we say that two polynomials $p(x)$ and $q(x)$ are apolar when $A(q(x), p(x))$ is identically zero, that is, zero for all $x$. Theorem 1 remains valid as stated. That is, if $q(r) = 0$ then the polynomial $p(x) = (x-r)^n $ is apolar to $q(x)$. Let us consider a special case. Suppose that $q(x)$ is a quadratic polynomial and $p(x)$ is a cubic polynomial: $$ q(x) = x^2 + 2b_1 x + b_2 $$ and $$ p(x) = x^3 + 3 a_1 x^2 + 3a_2 x + a_3 .$$ Then we have, in umbral notation $$ A(q(x), p(x)) \cong (\beta - \alpha)^2 (x - \alpha) = \\ (\beta^2 - 2 \alpha \beta + \alpha^2)x - \alpha \beta^2 + 2 \alpha^2 \beta - \alpha^2. $$ Evaluating the linear functional $E$, we obtain the following explicit expression for the apolar invariant: $$ A(q(x), p(x)) = E((\beta^2 - 2 \alpha \beta + \alpha^2)x - \alpha \beta^2 + 2 \alpha^2 \beta - \alpha^2) =$$ $$ E (\beta^2 ) - 2E( \alpha \beta) +E( \alpha^2)x +E ( - \alpha \beta^2) + 2 E(\alpha^2 \beta) -E( \alpha^2) =$$ $$E (\beta^2 ) - 2E( \alpha)E( \beta) +E( \alpha^2)x +E ( - \alpha)E( \beta^2) + 2 E(\alpha^2)E( \beta) -E( \alpha^2) = $$ $$ (b_2 - 2 a_1b_1 + a_2)x - a_1 b_2 + 2 a_2 b_1 - a_3. $$ Thus, a quadratic polynomial $q(x)$ and a cubic polynomial $p(x)$ are apolar if and only if their coefficients satisfy the two equations $$ b_2 - 2 a_1b_1 + a_2 = 0 $$ $$ - a_1 b_2 + 2 a_2 b_1 - a_3 = 0 . $$ Using these equations, we can prove two important theorems: \bigskip {\bf Theorem 2}. There is in general one quadratic polynomial which is apolar to a given cubic polynomial. \bigskip Proof. Indeed, the above equations may be rewritten as $$ b_2 - 2 a_1b_1 = - a_2 $$ $$ - a_1 b_2 + 2 a_2 b_1 = a_3. $$ The solutions $b_1, b_2$ for given $a_1, a_2, a_3 $ is in general unique. \bigskip {\bf Theorem 3}. There is always a two-dimensional space of cubic polynomials which are apolar to a given quadratic polynomial. \bigskip Proof. Indeed, given $b_1, b_2$ we may solve for $a_1, a_2, a_3 $ from the equations $$- 2 a_1b_1 + a_2 = - b_2 $$ $$ - a_1 b_2 + 2 a_2 b_1 = a_3 .$$ These equations always have a double infinity of solutions, as they used to say in the old days. \bigskip Theorems 2 and 3 provide a simple and explicit method for solving a cubic equation. It goes as follows. Given the cubic polynomial $$ p(x) = x^3 + 3 a_1 x^2 + 3a_2 x + a_3 ,$$ first, by Theorem 2 we find a unique quadratic polynomial $q(x)$ which is apolar to $p(x)$. In general, such a quadratic polynomial $q(x)$ has two distinct roots $r_1$ and $r_2$. By Theorem 1, the cubic polynomials $ (x-r_1)^3$ and $(x-r_2)^3$ are apolar to $q(x)$. Second, by Theorem 3, the affine linear space of cubic polynomials apolar to $q(x)$ has dimension two. Since $p(x)$ is apolar to $q(x)$, we conclude that $p(x)$ is a linear combination of $ (x- r_1)^3$ and $(x-r_2)^3$. In symbols: $$ p(x) = c (x-r_1)^3 + (1-c) (x-r_2)^3 $$ for some constant $c$. Observe that $c$, $r_2$ and $r_2$ are computed by solving linear and quadratic equations. In this way, the solution of the cubic equation $p(x) = 0$ is reduced to the solution of the equation $$ c (x-r_1)^3 = - (1-c) (x-r_2)^3 ,$$ and this equation is easily solved by taking a cube root. \bigskip This method of solving a cubic equation is the only one I can remember. \bigskip Let me digress with a personal anecdote. A few years ago, I was lecturing on this material at a symposium in combinatorics that took place at the University at Minnesota. Persi Diaconis was sitting in the front row, and I could tell as I started to lecture that he was falling asleep; he eventually began to doze off. But the moment I mentioned the magic words "solving a cubic equation" he woke up with a start and said: "Really! How?" \bigskip The preceding two theorems are easily generalized. \bigskip {\bf Theorem 4}. The dimension of the space of all (monic) polynomials of degree $k$ which are apolar to a polynomial of degree $n$ equals $2k-n$, in general. \bigskip {\bf Theorem 5}. The dimension of the space of all (monic) polynomials of degree $n$ which are apolar to a polynomial of degree $k$ equals $k$. \bigskip Let us try to solve an equation of degree 5 in much the same way as we solved a cubic equation. Given the quintic polynomial $$ p(x) = x^5 + 5a_1x^4 + 10 a_2 x^3 + 10 a_3 x^2 + 5 a_4 x + a_5 = 0 ,$$ Theorem 4 assures us that there is in general a unique cubic polynomial $q(x)$ which is apolar to $p(x)$. In general, this cubic polynomial has three distinct roots $r_1, r_2, r_3$. By Theorem 1, the polynomials $(x-r_1)^5, (x-r_2)^5, (x-r_3)^5 $ are linearly independent and apolar to $q(x)$ . By Theorem 5, the dimension of the space of all polynomials apolar to $q(x)$ equals $3$. But the polynomial $p(x)$ is apolar to $q(x)$. Hence, $p(x)$ can be written in the form $$ p(x) = c_1(x-r_1)^5 + c_2(x-r_2)^5 + c_3(x - r_r)^5 $$ for suitable constants $c_i$. Thus, we see that a generic polynomial of degree 5 can be written as a linear combination of three fifth powers of linear polynomials. These are computed by solving linear, quadratic and cubic equations. This reduction to canonical form of the quintic is as close as we can come to solving a quintic equation by radicals. \bigskip At this point, someone in the audience will raise his or her hand and say: "Excuse me, but the umbral method you have introduced is not even good enough to express the discriminant of a quadratic equation! " Quite right. The definitions of umbrae and of the linear functional $E$ have an obvious generalization to any array of polynomials, say $p_1(x), p_2(x),\dots , p_{\ell}(x) $. One simply considers the space of polynomials $$ {\bf C}[x, \alpha_1 ,\alpha_2, \dots , \alpha_{\ell} ] $$ and one sets $$E(\alpha_{t}^j )$$ to equal the $j$-th coefficient of the polynomial $p_{t}(x)$. What is crucial, the linear functional $E$ is again multiplicative on distinct umbrae: $$ E(\alpha_1^i \alpha_2^j \alpha_3^k \dots x^{\ell} ) = E(\alpha_1^i )E(\alpha_2^j) E( \alpha_3^k )\dots x^{\ell}.$$ Now comes the catch that in the old days was turn into a notational nightmare: the polynomials say $p_1(x), p_2(x),\dots , p_{\ell}(x) $ need not be distinct. In fact, the most important case occurs when each of the polynomials $p_1(x), p_2(x),\dots , p_{\ell}(x) $ is equal to one and the same polynomial $p(x)$. In this case, the definition of the linear functional $E$ may be simplified as follows: 1. $$E(\alpha_i^j) = a_j $$ for every $i$ , and 2. $$ E(\alpha_1^i \alpha_2^j \alpha_3^k \dots x^{\ell}) = a_i a_j a_k \dots x^{\ell} $$ for all non negative integers $i,j, k, \dots \ell $. \bigskip Umbrae $ \alpha_1 ,\alpha_2, \dots , \alpha_{\ell} $ satisfying 1 and 2 are said to be exchangeable. Thus, for exchangeable umbrae we have $$ (x + \alpha_1)^n \cong (x+\alpha_2)^n. $$ Eric Temple Bell, who wrote $=$ in place of $\cong$, was baffled by the fact that two umbrae could be be exchangeable without being equal. \bigskip We can now state the main theorem of invariant theory. We will consider a single polynomial. \bigskip {\bf Theorem 5}. Every invariant of a polynomial $p(x)$ is obtained by evaluating some polynomial in the differences $\alpha_i - \alpha_j $ and $\alpha_i - x,$ where $\alpha_i $ and $\alpha_j $ are exchangeable umbrae. Conversely, every polynomial in such differences is equivalent to an invariant of the polynomial $p(x)$. \bigskip The proof is extremely simple, but will be omitted. \bigskip Let us review some classical examples. \bigskip The discriminant of a quadratic polynomial $p(x) = x^2 +2a_1x+a_2 $ may be umbrally represented as follows: $$ D(p(x)) \cong (\alpha_1 - \alpha_2)^2 / 2, $$ where $\alpha_1 $ and $\alpha_2 $ are exchangeable umbrae. Indeed: $$ E( (\alpha_1 - \alpha_2)^2 ) =E( \alpha_1^2) - 2 E(\alpha_1\alpha_2) + E(\alpha_2 ^2) =$$ $$ E( \alpha_1^2) - 2 E(\alpha_1)E(\alpha_2) + E(\alpha_2 ^2) =a_2 - 2 a_1^2 + a_2 = 2 (a_2 - a_1^2), $$ as desired. \bigskip Let us next consider a cubic polynomial $p(x) = x^3 + 3a_1 x^2 + 3a_2 x + a_3 $. The discriminant of this polynomial, let us call it $D(p(x)) $, equals, as you know, the expression $$ D(p(x)) = \frac{4(a_2 - a_1)^2)(a_1a_3 - a_2^2) - (a_3 - a_1a_2)^2}{2} .$$ The umbral expression of the discriminant is easier to remember. $$ D(p(x)) \cong (\alpha_1 - \alpha_2)^2(\alpha_3 - \alpha_4)^2(\alpha_1 - \alpha_4)(\alpha_2 - \alpha_3). $$ As you know, the discriminant vanishes if and only if the cubic equation $p(x) = 0 $ has a double root. \bigskip The Hessian of a cubic polynomial can be elegantly written in umbral notation as follows: $$ H(p(x)) \cong (\alpha_1 - \alpha_2)^2(\alpha_1 - x)(\alpha_2 - x). $$ The Hessian vanishes if and only if the cubic polynomial is the third power of a polynomial of degree one. \bigskip Allow me another digression. On hearing about the vanishing of the Hessian as the condition that a cubic polynomial be a perfect cube, it comes naturally to ask the general question: which invariant of a polynomial of degree $n$ vanishes if and only if the polynomial is the $k$- th power of some polynomial of degree $n/k$? Here $k$ is a divisor of $n$. For a long time I thought the answer to this question to be beyond reach, until one day, while leafing despondently through the second volume of Hilbert's collected papers, I accidentally discovered that Hilbert had completely solved it. The solution can be elegantly expressed in umbral notation. This is only one of several striking results of Hilbert's on invariant theory that have been forgotten. \bigskip Let us consider next an invariant of the quintic. Theorem 3 tells us that a quintic $p(x) = x^5 + 5a_1x^4 + 10 a_2x^3 + 5 a_3x^4 + a_5 $ has a unique apolar cubic polynomial $q(x)$. The polynomial $q(x)$ is an invariant of $p(x)$. Does it have a simple expression in umbral notation? Indeed it does. The expression is the following: $$ q(x) \cong (\alpha_2 - \alpha_3)^2(\alpha_3 - \alpha_1)^2(\alpha_1 - \alpha_2)^2(\alpha_1 - x)(\alpha_2 - x)(\alpha_3 - x). $$ In the classical literature this invariant is denoted by the letter $j$. What property will the quintic polynomial $p(x)$ have when the invariant $j$ vanishes? The answer to this question is pleasing. The invariant $j$ of a quintic polynomial is identically equal to zero if and only if the quintic is apolar to some non trivial polynomial of degree two. But then Theorem 3 tells us that the quintic may be written in the form $$ p(x) = c(x-r_1)^5 + (1-c)(x-r_2)^5, $$ where $r_1, r_2 $ are the roots of a quadratic equation. Thus, the vanishing of the invariant $j$ is a necessary and sufficient condition that the quintic polynomial $p(x)$ may be written as the sum of two rather than three fifth powers of linear polynomials. When this is the case, the fifth degree equation $p(x) = 0$ can be solved by radicals. By similar arguments, one can compute all invariants whose vanishing implies that the equation of degree five is algorithmically solvable by radicals. Twenty-three invariants play a relevant role, as Cayley was first to show. \bigskip Hilbert's theorem on finite generation of the ring of invariants can be recast in the language of umbrae, and can be given a simple combinatorial proof that dispenses with the Hilbert basis theorem. \bigskip In closing, let us touch opon another reason for the demise of the symbolic method in invariant theory. \bigskip In mathematics, it is extremely difficult to tell the truth. The formal exposition of a mathematical theory does not tell the whole truth . The truth of a mathematical theory is more likely to be grasped while we listen to a casual remark made by some expert that gives away some hidden motivation, when we finally pin down the typical examples, or when we discover what the real problems are that were stored behind the showcase problems. \bigskip Philosophers and psychiatrists should explain why it is that we mathematicians are in the habit of systematically erasing our footsteps. Scientists have always viewed askance at this strange habit of mathematicians, which has changed little from Pythagoras to our day. \bigskip The hidden purpose of the symbolic method in invariant theory was not simply that of finding easy expressions for invariants. A deeper faith was guiding this method. It was the expectation that the expression of invariants by the symbolic method would eventually guide us to single out the "relevant" or "important" invariants among an infinite variey. It was the hope that the significance of the ending of an invariant could be gleaned from its umbral expression. The vanishing of this faith was the real reason for the demise of classical invariant theory, and the revival of this faith is the reason for its present rebirth. \bigskip Whether or not we will succeed this second time where the classics failed is a cliffhanger that will probably be resolved in the next few years. I would not be speaking to you now if I did not believe in success. \bigskip Thank you for your attention. \newpage \begin{center} {\bf COMBINATORIAL SNAPSHOTS\\ being\\ The third of three Colloqium Lectures\\ delivered at the Annual Meeting of the American Mathematical Society\\ Baltimore, January 9, 1998\\ Gian-Carlo Rota\\ Department of Mathematics\\ MIT\\ Cambridge MA 02139-4307\\} \end{center} \bigskip \bigskip When I was in high school, my English teacher gave me to read an essay by James Thurber, called "The secret life of Walter Mitty". After rereading this essay every few years, I decided that everyone has a Walter Mitty complex. One way to understand a person might be to discover that person's Walter Mitty fantasies. Most of our mathematical thoughts in high school or in college were Walter Mitty fantasies. When we learned a new piece of math, we would find ourselves fantasizing on its possible generalizations. As soon as we understood binomial coefficients, we fantasized about their generalization to the case when the denominator is negative; the moment we learned about derivatives, we launched into derivatives of fractional order. If we were ever exposed to the Riemann zeta function, we would romanticize some new interpretation of this function that would give away its secret. \bigskip This lecture should have been given another title. It should be called "The later life of Walter Mitty". It will consist of a sequence of displays of "chutzpah" by a Walter Mitty who has lost his shyness. Each snapshot will deal with the realization of some of youthful fantasies that have partially worked out. \bigskip FIRST SNAPSHOT: AN EXAMPLE OF PROFINITE COMBINATORICS. \bigskip Let us begin with a piece of history-fiction, and fantasize how Riemann might have discovered the Riemann zeta function. \bigskip Professor Riemann was aware that arithmetic density is of fundamental importance in number theory. If $A$ is a subset of the set of set of positive integers $N$, then the arithmetic density of the set $A$ is defined to be $$ dens(A)=\lim_{n\rightarrow \infty} {1\over n} |{A\cap \{1,2, \dots , n \}|.$$ \bigskip whenever the limit exists. For example $dens(N) = 1$. If $A_p$ is the set of multiples of the prime $p$, then $dens(A_p) = \frac{1}{p}$; what is more appealing, one easily computes that $dens (A_p \cap A_q) = \frac{1}{pq} $ for any two primes $p$ and $q$. If density were a (countably additive) probability measure, we would infer that the events that a randomly chosen number is divisible by either of two primes are independent. Unfortunately, arithmetic density shares some but not all properties of a probability measure. It is most emphatically not countably additive. \bigskip After a period of soul searching, Professor Riemann was able to find a remedy to some deficiencies of arithmetic density by a brilliant leap of imagination. He chose a real number $s > 1$, and defined the measure of a positive integer $n$ to equal $\frac{1}{n^s}$; in this way, the measure of the set $N$ turned out to equal $$ \zeta (s) = \sum_{n=1}^{\infty}\frac{1}{n^s}. $$ \bigskip Therefore, he could define a (countably additive) probability measure $P_s$ on the set $N$ of positive integers by setting $$ P_s(A) = \frac{1}{\zeta (s)} \sum_{n \in A }\frac{1}{n^s}. $$ Riemann then proceeded to verify what he had sensed all along, namely, the fundamental property $$P_s(A_p \cap A_q) = P_s(A_p)P_s(A_q) = \frac{1}{pq}. $$ In other words, events $A_p$ and $A_q$ that a randomly chosen integer $n$ is divisible by either of two primes $p$ or $q$ are independent relative to the probability $P_s $. The Riemann zeta function was good for something, after all. \bigskip I will now use a rhetorical device that was effectively employed by one my undergraduate teachers, Professor Bochner. In the classroom, Professor Bochner would prefix the statement of a theorem by the words: "Subject to technical assumptions, the following is true"; without, of course, ever disclosing what his technical assumptions were. \bigskip Professor Riemann then proceeded to show that, subject to techincal assumptions on the set $A$, $$ \lim_{s\rightarrow 1}P_s(A) = dens (A). $$ \bigskip Thus, even though arithmetic density is not a probability, it is under suitable conditions the limit of probabilities. \bigskip Long after Riemann was gone, it was shown, again subject to technical assumptions, that the probabilities $P_s$ are the only probabilities defined on the set $N$ of natural integers for which the events of divisibility by different primes are independent. This fact seems to lend support to the program of proving results of number theory by probabilistic methods based upon the Riemann zeta function. \bigskip Why didn't Professor Riemann ever publish this wonderful idea of his? The answer is not hard to find. True, some theorems of number theory can be proved probabilistically by this limiting process, for example Dirichlet's theorem on primes in arithmetic progression. However, deeper number theoretic results have to this day eluded this approach, for example, no one has succeeded in proving the prime number theorem by this method. Professor Riemann, aware of this deficiency, threw his notes into the wastebasket and proceeded to link the Riemann zeta function to the distribution of primes in an altogether different way, by stating the hypothesis that bears his name and that remains unproved to this day. \bigskip Why am I telling you this bit of history-fiction? Because I want to propose another probabilistic interpretation of the Riemann zeta function that is quite different from the interpretation just outlined. \bigskip Let us consider a problem in combinatorial enumeration. Given a cyclic group of order $r$, say $C_r $. Every character $\chi $ of the group $C_r$ has a kernel which is a subgroup of $C_r$. More generally, every sequence $\chi_1, \chi_2, \dots , \chi_s $ of characters of $C_r$ has a joint kernel which is also a subgroup of $C_r$; the joint kernel of a sequence of characters is simply the intersection of the kernels of each of the characters in the sequence $\chi_1, \chi_2, \dots , \chi_s $ . If a sequence $\chi_1, \chi_2, \dots , \chi_s $ of $s$ characters is chosen independently and at random, what is the probability that the joint kernel of the sequence equals a given subgroup $C_n $ of $C_r $? \bigskip The probability of the event that the the kernel of a randomly chosen character will contain the subgroup $C_n$ equals $\frac{1}{n}$, since there are $r$ characters of the group $C_r$ and $\frac{r}{n}$ such characters will vanish on $C_n $. Therefore, the probability that the joint kernel of a randomly chosen sequence $\chi_1, \chi_2, \dots , \chi_s $ of $s $ characters shall contain the subgroup $C_n $ equals $(\frac{1}{n})^s $. Let us denote by $P_{C_n}$ the probability that the joint kernel of the characters $\chi_1, \chi_2, \dots , \chi_s $ shall equal the subgroup $C_n $. Then we have the identity $$ { 1 \over n^s} = \sum_{n|d|r} P_{C_d}.$$ Here, we use the fact that the partially ordered set of subgroups of a cyclic group $C_r$ is isomorphic to the partially ordered set of divisors of the integer $r$. We now use the M\"obius inversion formula of number theory, thereby obtaining $$ P_{C_n} = \sum_{n|d|r} \mu (d/n){1 \over d^s}.$$ Here, $\mu(j) $ is the M\"obius function of number theory. After the change of variable $d = nj $ we can recast the right hand side as follows: $$ P_{C_n} = { 1 \over n^s} \sum_{j}\mu (j){1 \over j^s}.$$ The variable $j$ on the right ranges over some subset of divisors of the integer $r$, which we need not worry about. \bigskip Now if the sum on the right ranged over all positive integers $j$, then the right hand side would equal $${ 1 \over n^s} \frac{1}{\zeta(s)},$$ \bigskip that is, it could be expressed in terms of the inverse of the Riemann zeta function. If we could change our combinatorial problem to get an unrestricted sum on the right hand side, then we would have a probabilistic interpretation of the Riemann zeta function. \bigskip This is done by replacing the finite cyclic group $C_n$ by a profinite cyclic group. \bigskip Consider the group $C_{\infty}$ of rational numbers $mod 1 $. For every positive integer $n$, the group $C_{\infty}$ has a unique finite subgroup $C_n$ with $n$ elements. The character group $C_{\infty}^*$ of $C_{\infty}$ is a compact group; it has a Haar measure which is a probability measure $P$. The group $C_{\infty}^* $ is the desired profinite group on which we can generalize the preceding computation. \bigskip The set of all characters of the group $C_{\infty}$ (that is, the set of all elements of the group $C_{\infty}^* $) which vanish on a subgroup $C_n$ of $C_{\infty}$ has Haar measure equal to $\frac{1}{n}$. Thus, if we choose a sequence $\chi_1, \chi_2, \dots , \chi_s $ of $s $ characters of $C_{\infty}$ independently and at random, the probability that their joint kernel will contain the group $C_n$ equals $(\frac{1}{n})^s$ . If we again denote by $ P_{C_n} $ the probability that the joint kernel of a sequence $\chi_1, \chi_2, \dots , \chi_s $ of $s $ characters equals the group $C_n$, then we have the identity $${ 1 \over n^s} = \sum_{n|d} P_{C_d},$$ where the sum on the right is now infinite. Again by the M\"obius inversion formula we obtain $$ P_{C_n} = \sum_{n|d} \mu (d/n){ 1 \over d^s} = \frac{1}{n^s}\frac{1}{\zeta(s)}.$$ This is the promised probabilistic interpretation of the Riemann zeta function. Some properties of the Riemann zeta function can be proved probabilistically using this interpretation, for example, the product formula. It remains to be seen which other properties of the Riemann zeta function can be proved in this way. \bigskip The preceding argument is as an instance of a generalization of an enumeration problem on a finite set to an enumeration on a profinite set. Such a replacement of a finite set by a profinite "set" works in other combinatorial problems. Will we ever have a profinite combinatorics on profinite sets side by side with combinatorics on finite sets? \newpage SECOND SNAPSHOT: THE CYCLIC DERIVATIVE. \bigskip The ordinary derivative of a polynomial in one variable has been generalized by Hausdorff to polynomials and formal power series in non commutative variables as follows. Consider the associative algebra ${\bf{C}} \langle a,b, \dots , c , x \rangle $ generated by a set of letters $\{a, b, \dots, c, x\}$. The letter $x$ is called a variable, all other letters are called constants. A monomial in this associative algebra is what you think it should be: it is a word like $$ m = axbax^3bcxd.$$ A polynomial is a linear combination of monomials, and a formal power series is defined as an infinite sum of monomials, with suitable restrictions on the growth of degrees of the summands. Formal power series in non commutative variables form an algebra ${\bf C}\langle \langle a,b,\dots , c, x \rangle \rangle $. We will denote by $f(x)$ such a formal power series. The Hausdorff derivative of the monomial $m$ is computed as follows: $$ H (m) = H(axbax^3bcxd) = abax^3bcxd +3 axbax^2bcxd + axbax^3bcd. $$ This definition is extended by linearity to polynomials and to formal power series. \bigskip If $m'$ is another monomial, we have the expected rule for finding the Hausdorff derivative of a product: $H(m m') = H(m)m' + m H(m'). $ \bigskip The Hausdorff derivative suffers from a major weakness. There seems to be no analog of the chain rule for the differentiation of a function of a function. For example, the Hausdorff derivative of the polynomial $(ax)^n$, when the letters $a$ and $x $ do not commute, is not equal to $n (ax)^{n-1}a $. It is a mess. \bigskip There is another notion of derivative that does satisfy a simple chain rule under functional composition. It is the cyclic derivative, denoted by the letter $D$. \bigskip The cyclic derivative is defined as follows. First define the truncation operator $T$ as follows: \bigskip a. if the first letter of a monomial $m$ is not the variable $x$, set $T(m) = 0;$ \bigskip b. if the first letter of a monomial $m$ is the variable $x$, so that $m = x m' $, set $T(m) = m'. $ \bigskip c. Extend by linearity to ${\bf C}\langle \langle a,b,\dots , c, x \rangle \rangle $. \bigskip The cyclic derivative of a monomial $m$ is defined in terms of the truncation operator follows: \bigskip a. Let $p$ be the polynomial obtained by adding all cyclic permutations of the monomial $m$. \bigskip b. Set $D(m) = T(p).$ \bigskip c. Extend by linearity to all formal power series. \bigskip For example, the cyclic derivative of the above monomial $m$ is computed in the following steps: \bigskip Step 1. Write down all cyclic permutations on the monomial $ axbax^3bcxd$. These are $$ xbax^3bcxda , bax^3bcxdax , x^3bcxdaxba , x^2bcxdaxbax , xbcxdaxbax^2 ,$$ $$bcxdaxbax^3 , cxdaxbax^3b , xdaxbax^3bc , daxbax^3bcx . $$ \bigskip Step 2. In the above list, perform one of the following operations: \bigskip a. if the first letter of a monomial is not $x$, remove the monomial from the list; \bigskip b. if the first letter of a monomial is $x$, remove the first letter. \bigskip When we perform operations a. and b. on the each of the monomials in the above list, we obtain a shorter list, namely: $$ bax^3bcxda , x^2bcxdaxba , xbcxdaxbax, bcxdaxbax^2 , daxbax^3bc. $$ Step 3. Add the monomials thus obtained to get the cyclic derivative: \begin{eqnarray*}D(m) & = & D(axbax^3bcxd) \\ & = & bax^3bcxda + x^2bcxdaxba + xbcxdaxbax+ bcxdaxbax^2 + daxbax^3bc. \end{eqnarray*} Another example: the cyclic derivative of the monomial $axbxcxdx$ equals $$D(axbxcxdx) = bxcxdxa + cxdxaxb + dxaxbxcx + axbxcxd. $$ The cyclic derivative of the monomial $(ax)^n$ is the following: \begin{eqnarray*} D((ax)^n) & = & D(axax \dots ax)\\ & = & axax\dots axa + axax\dots axa + \dots axax\dots axa \\ & = & n (ax)^{n-1} a.\end{eqnarray*} Similarly, one computes $$ D(x+a)^n = n(x+a)^{n-1}$$ and, for formal power series, $$ D(e^{x+a}) = e^{x+a} $$ and $$ D(e^{ax}) = e^{ax}a. $$ Remember, the letters $a$ and $x $ do not commute! In these examples, the corresponding Hausdorff derivative is a mess. The cyclic derivative enjoys all properties expected of the ordinary derivative; in particular, it satisfies the chain rule for the composition of two formal power series. To state the rules for taking cyclic derivatives, we need one more operator, called the wrapping operator. The wrapping operator is defined as follows. Let $c_1,c_2,\dots ,c_n $ be any letters. If $g(x)$ is any formal power series, set $$ \langle C c_1c_2\dots c_n | g(x) \rangle\\ = $$ c_1c_2\dots c_n g(x) + c_2\dots c_n g(x)c_1 + c_3 \dots c_n g(x)c_1c_2 + \dots + c_n g(x)c_1c_2\dots c_{n-1}.$$ \bigskip If $f(x)$ is any formal power series, the wrapping operator $$ \langle C f(x) | g(x) \rangle $$ is defined by linearity. Set $$ \langle D(f(x)) | g(x) \rangle = T \langle C f(x) | g(x) \rangle. $$ For example: $$ \langle D(f(x)) | 1 \rangle = D(f(x)). $$ \bigskip The cyclic derivative of the product of two "functions"is given by the following identity: $$ D(f(x)g(x)) = \langle D(f(x)) | g(x) \rangle + \langle D(g(x)) | f(x) \rangle. $$ For example, one obtains \bigskip $$ D((1-ax)^{-1}(1-bx)^{-1}) & = & (1-ax)^{-1}(1-bx)^{-1}(1-ax)^{-1}a $$ $$ +(1-bx)^{-1}(1-ax)^{-1}(1-bx)^{-1}b .$$ \bigskip No such identity holds for the Hausdorff derivative. The cyclic derivative of the product of any sequence of formal power series is similarly computed by the wrapping operator: $$D(f_1(x)f_2(x)\dots f_n(x)) = $$ $$\langle D(f_1(x)) | f_2(x) \dots f_n(x) \rangle + \langle D(f_2(x)) | f_3(x) \dots f_n(x)f_1(x) \rangle + $$ $$ \dots\\ + \langle D(f_n(x)) | f_1(x)f_2(x) \dots f_{n-1}(x) \rangle. $$ \bigskip We come now to the main property of the cyclic derivative: the chain rule. Given two formal power series $f(x) $ and $g(x) $ in ${\bf C}\langle \langle a,b,\dots , c, x \rangle \rangle $, assume that formal power series $g(x) $ does not have a constant term. Under these circumstances, the composition $f(g(x))$ is well defined by replacing $g(x) $ for every occurrence of the variable $x$ in the formal power series $f(x)$. Let us write $ D_g(f(x)) $ to denote the formal power series obtained by substituting $g(x)$ in place of every occurrence of $x$ in the cyclic derivative $D(f(x))$ of the formal power series $f(x)$. Then the chain rule for the cyclic derivative goes as follows: $$ D(f(g(x))) = \langle Dg(x) | D_g(f(x)) \rangle. $$ For example, we have $$ D(e^{axbx}) = bx e^{axbx} a + e^{axbx}axb. $$ A more elegant example is the following: $$ D(e^{(1-ax)^{-1}}) = (1-ax)^{-1} e^{(1-ax)^{-1}}(1-ax)^{-1} a. $$ One can prove that the cyclic derivative of a rational formal power series in non commutative letters is again a rational non commutative power series , and that the cyclic derivative of an algebraic formal power series in non commutative letters is again an algebraic formal power series. Despite the evidence that the cyclic derivative is the natural notion of devivative for non commutative algebras, the theory as it is at present is not satisfying. The cyclic derivative is an empirical discovery. It needs to be ensconced in some broader algebraic theory, much like the Hausdorff derivative has been ensconced in the theory of Hopf algebras. \newpage THIRD SNAPSHOT: LOGARITHMS AND THE BINOMIAL THEOREM. \bigskip The Euler-MacLaurin summation formula is one of the most remarkable formulas of mathematics. For a suitable function $f(x)$ of a real or complex variable, it is stated as follows: $$ f(x) + f(x+1) + f(x+2) + \dots + f(x+n) =$$ $$ B_0 \int_x^{x+n+1}f(y ) dy + B_1 (f(x+n+1) - f(x)) $$ $$ + {B_2 \over 2!} D (f(x+n+1) - f(x)) + {B_3 \over 3!} D^2 (f(x+n+1)- f(x)) + \dots .$$ The $B_n$ are the Bernoulli numbers and where $D$ is the ordinary derivative operator. The Euler-MacLaurin formula has proved very useful for over two hundred years. Nonetheless, the Euler-MacLaurin formula suffers from a serious deficiency. The series on the right hand side is almost never convergent, unless it reduces to a finite sum. Our question is the following: is there a vector space of functions which contains as many of the elementary functions as possible, and a topology on such a vector space, relative to which the right hand side of the Euler-MacLaurin formula is a convergent series? The answer to this question is unexpectedly related to the answer to another question. What is the "right" generalization of the binomial coefficients ${n \choose k}$ when $k$ is allowed to be a negative integer? This question leads in turn to a third question: how shall we know whether a generalization of the binomial coefficients is "right"? The answer to this third question is easy: a generalization of the binomial coefficients is "right" if it leads to a sensible generalization of the binomial theorem: \bigskip $$ (a+ x)^n = \sum_{k=0}^n {n \choose k}a^k x^{n-k} $$ \bigskip When I was young, I used to think of the binomial theorem as trivial. I think I have learned my lesson. A well-known philosopher, I can't remember his name, wrote that the whole universe can be inferred from a grain of sand. He should have added that a great deal of mathematics can be derived by meditating upon the binomial theorem. \bigskip Let us take the bull by the horns, and state the "right" generalization of the binomial coefficients. We proceed in the most pedestrian way, by first generalizing the definition of the factorial. Thus, let $n$ be any integer, positive or negative. We define the Roman factorial $[n]!$ as follows: $$ [n]! = n! $$ if $ n \geq 0 $, and $$ [n]! = {(-1)^{n+1} \over (-n-1)!} $$ if $ n < 0 $. Where does this definition come from ? I could simply say that it works, but that would not be the whole truth. The value of the Roman factorial $ [n]! $ for $n$ negative equals the residue of the gamma function at the integer $n$. Using the Roman factorial, we define the Roman coefficients as follows: $$ { n \brack k} = { [n]! \over [k]![n-k]!}. $$ When $ n \geq k \geq 0 $, the Roman coefficients coincide with the binomial coefficients. For all integers $n$ and $k$, the Roman coefficients share all elementary properties of binomial coefficients, such as Pascal's triangle, etc. However, there are some surprises in store, for example, for $k$ positive we find $$ {0 \brack -k}= {0 \brack k} = { (-1)^{k+1} \over k }. $$ \bigskip Does this make any sense? Well, yes, because we can find a generalization of the binomial theorem that goes with this. It is the following. Recall the power series expansion of the logarithm: $$ log(x+a) = log x + \sum_{k=1}^{\infty}{ (-1)^{k+1} \over k} {a^k \over x^k}.$$ We can recast this power series expansion in terms of the Roman coefficients as follows: $$ log(x+a) = log x + \sum_{k=1}^{\infty} {0 \brack k} {a^k \over x^k}. $$ This is beginning to look like a generalization of the binomial theorem, with the logarithm playing the roles of zero-th power. Another power series expansion where the Roman coefficients make their appearance is the following: $$ (x+a)(log(x+a) - 1) = x(log x - 1) + a log x + \sum_{k=2}^{\infty}{1 \brack k} a^k x^{1 -k}. $$ Do we see a pattern? Well, let us try yet another power series expansion: $$ (x +a)^2(log(x+a) - 1 - {1 \over 2}) =$$ $$ x(log x - 1 - {1 \over 2}) + {2 \brack 1}a x(log x - 1) + {2 \brack 2}a^2 log x + \sum_{k=3}^{\infty} {2 \brack k} a^k x^{2-k}. $$ Now we can leap to a generalization. For suitable functions $f(x)$, set $$ D^{-1}f(x) $$ to be the unique indefinite integral of the function $f(x)$ which has constant term equal to zero. Do not worry, this will make sense in a moment. \bigskip Define $$ \lambda_n^{(1)} (x) = [n]! D^{-n}log x. $$ Here, $n$ is any integer, positive or negative. The functions $\lambda_n^{(1)}(x)$ are called the harmonic logarithms of order one. For $n$ positive we have $$ \lambda_n^{(1)} (x) = x^n(log x - 1 - {1\over 2} - {1 \over 3} - \dots -{1 \over n}) $$ and $$ \lambda_{-n}^{(1)} (x) = {1 \over x^n}.$$ Of course we also have $$ \lambda_0^{(1)} (x) = log x.$$ We are now in a position to state the generalization of the binomial theorem that is associated with the harmonic logarithms. It goes as follows: $$ \lambda_n^{(1)} (x+ a) = \sum_{k=0}^{\infty}{n \brack k} a^k \lambda_{n-k}^{(1)} (x). $$ The three identities above are special cases of this identity, for $n = 0,1,2$. The generalization of the binomial theorem to harmonic logarithms gives nothing new for negative exponents, where it reduces to the identity $$(x+a)^{-n} = \sum_{k=0}^{\infty} { -n \choose k} a^k x^{-n-k}.$$ However, for positive exponents we obtain a genuine and baffling generalization of the binomial theorem. It states that the functions $ \lambda_n^{(1)} (x) $, for $n $ positive, satisfy the ordinary binomial theorem, modulo negative powers of $x$. In other words, we have the following identity: $$ (x+a)^n(log(x+a) - 1 - {1\over 2} - {1 \over 3} - \dots - {1 \over n}) \cong $$ $$\sum_{k=0}^n {n \choose k}a^k x^{n-k}(log(x) - 1 - {1\over 2} - {1 \over 3} - \dots - {1 \over {n-k}}). $$ The identity is valid modulo negative powers of $x$. Miracles of cancellation are occurring in this identity. I wish I knew a combinatorial or probabilistic interpretation of this logarithmic generalization of the binomial theorem. \bigskip So far, we have assumed that all series converge in the topology of the complex numbers. We will now change the topology, while retaining the convergence. The motivation for the logarithmic topology we are about to define is the algebra of formal Laurent series. This topological algebra may be defined by defining a topology on the algebra of rational functions in the variable $x$, and then completing this algebra relative to the topology. The topology is so chosen as to have $lim_{n \rightarrow \infty}x^{-n} = 0 $. Every element of the completed algebra turns out to be a formal Laurent series, that is, a series of the form $$ \sum_{n < d} a_n x^n. $$ We want to perform an analogous completion process on another algebra: the algebra generated by all functions of the form $x^n(logx)^t$, where $n$ is any integer, positive or negative, and where $t$ is a non negative integer. In order to specify which elements of this algebra are to converge to zero, we need a better behaved basis of this algebra. This basis is provided by the harmonic logarithms of arbitrary order $t$. They are defined as follows: $$\lambda_n^{(t)}(x) = D^n (log x)^t $$ for every non negative integer $t$ and for every integer $n$ . For example, we have $$\lambda_n^{(0)}(x) = x^n $$ for every non negative integer $n$, and $$ \lambda_n^{(0)}(x) = 0 $$ for negative $n$. Explicit expressions are known for the harmonic logarithms. For the harmonic logarithms of order $2$ we have $\lambda_0^{(2)}(x) = (logx)^2 $ and for $n$ positive $$\lambda_n^{(2)}(x) = x^n \big( (logx)^2 - ( 2 + {2 \over 2} + \dots + {2 \over n})log x + 2 + {2 \over 2} (1+ {1 \over 2}) + \dots + {2 \over n}(1+ {1 \over 2} + \dots + {1 \over n}) \big), $$ and $$ \lambda_{-n}^{(2)}(x) = 2x^{-n} \big( log x - 1 - {1\over 2} - \dots - {1 \over n-1} \big). $$ For every non negative integer $t$ the harmonic logarithms of order $t$ satisfy the same generalization of the binomial theorem that we have already seen for the harmonic logarithms of order $1$: $$ \lambda_n^{(t)} (x+ a) = \sum_{k=0}^{\infty}{n \brack k} a^k \lambda_{n-k}^{(t)} (x). $$ The harmonic logarithms are a basis of the algebra generated by all functions $x^n(logx)^t$. We define a topology on this algebra by requiring that $$ lim_{n \rightarrow - \infty}\lambda_n^{(t)}(x) = 0 $$ for every non negative integer $t$. This topology is called the logarithmic topology. The completion of this algebra relative to the logarithmic topology is the algebra of formal power series of logarithmic type, or logarithmic algebra. \bigskip Every element of the logarithmic algebra is a linear combination of convergent power series of the form $$ f(x) = \sum_{n \leq d} b_n \lambda_n^{(t)}(x) $$ ranging over a finite set of values of $t$. bigskip We can now return to the Euler-MacLaurin summation formula: \bigskip {\bf Theorem}. For every element $f(x)$ of the logarithmic algebra the right hand side of the Euler-MacLaurin series converges in the logarithmic topology. \bigskip For example, the following infinite series is convergent in the logarithmic topology: $$ log x + log(x+1) + log(x+2) + \dots + log(x+n) = $$ $$B_0 ((x+n+1)log(x+n+1) - xlogx - n - 1 ) + B_1 ( log(x+n+1) - logx ) +$$ $${B_2 \over 2!} ({1 \over x+n+1 } - {1 \over x}) + \dots ) . $$ \bigskip Another example is the following. As you know, the sum $$ x^k + (x+1)^k + (x+2)^k + \dots + (x+n)^k $$ can be expressed in closed form by the Euler-Maclaurin formula. The preceding theorem leads to analogous closed form expressions for sums of the form $$ x^klogx + (x+1)^klog(x+1) + (x+2)^klog(x+2)+ \dots + (x+n)^klog(x+n) .$$ The harmonic logarithms have other applications, let us mention one in closing. \bigskip Recall the definition of the shift operator of the calculus of finite differences: $$E^af(x) = f(x+a).$$ For $n$ a non negative integer, define the operator $E_1$ as follows: $$ E_1 \lambda_n^{(0)}(x) = \lambda_n^{(1)}(x). $$ In ordinary notation, this is the same as saying $$ E_1 x^n = x^n(log x - 1 - 1/2 - 1/3/ - \dots - 1/n). $$ One can prove the following two propositions: \bigskip {\bf Proposition}. The operators $E^a$ and $E_1$ commute. \bigskip {\bf Proposition}. The restriction of the derivative operator $D$ to the subalgebra of the logarithmic algebra generated by the harmonic logarithms $\lambda_n^{(t)}(x)$ for positive $t$ (that is, excluding the non negative powers of $x$) is invertible. \bigskip These two propositions can be used to obtain "logarithmic extensions" of special functions. Let us conclude with the simplest example: let us compute the logarithmic extension of the sequence of lower factorials, namely, the polynomials $ (x)_n = x(x-1)(x-2)\dots (x-n+1)$. This sequence of polynomials satisfies the difference quation $$ \Delta (x)_n = n(x)_{n-1}.$$ This sequence can be extended to negative $n$ by setting $$ (x)_{-n} = { 1 \over {(x+1)(x+2)...(x+n)} }, $$ and we have $$ \Delta (x)_{-n} = -n(x)_{-n-1}.$$ For positive $n$, we may define the logarithmic extension of this sequence by setting $$ (x)_{-n}^{(1)} ={ 1 \over {(x+1)(x+2)...(x+n)} }. $$ For example, $(x)_{-1}^{(1)} = { 1\over x+1}$. The elements $ (x)_{-n}^{(1)}$ belong to the submodule of the logarithmic algebra spanned by $\lambda_n^{(1)}(x)$, as $n $ ranges over all integers. On this submodule, the operator $\Delta$ is invertible, and we can therefore set $$ (x)_n^{(1)} = \Delta^{-n-1}{1 \over x+1}$$ for all non negative integers $n$. It turns out that the element $(x)_0^{(1)}$ is given by the following series, convergent in the logarithmic topology: $$ (x)_0^{(1)} = log(x+1) + {B_1 \over 1+x} - {B_2 \over 2(1+x)^2} + {B_3 \over { 3(1+x)^3 }} - \dots . $$ But this is a familiar object: it is the $\Psi$- function, heuristically introduced by Gauss. Gauss motivated the $\Psi$-function as the "right" solution of the difference equation $$ \Delta \Psi(x+1) = {1 \over { x+1}}. $$ We have now rigorously verified Gauss's guess. Further computations show that the elements $ (x)_1^{(1)}$ and $ (x)_2^{(1)} $ also coincide with special functions introduced by Gauss, namely, the digamma and trigamma functions, which are at last rigorously defined by infinite series convergent in the logarithmic topology. In a similar vein, one defines logarithmic extensions of the Bernoulli polynomials, the Hermite polynomials, etc., and one finds that the asymptotic expansions of these polynomials reappear naturally as members of the logarithmic extensions of these functions. As a matter of fact, the logarithmic topology allows us to replace asymptotic expansions by series which are convergent in the logarithmic topology. \bigskip In closing, two open problems may be mentioned. First, no closed form expression is known for the coefficients of the expansion of a product $$\lambda_n^{(t)}(x) \lambda_k^{(s)}(x) $$ into a logarithmic power series. Second, we do not know a combinatorial or probabilistic interpretation of the Roman coefficients ${n \brack k}$ in general. \bigskip Thank you for listening. \end{document} --============_-1324990680==_============-- From sara(at-sign)math.mit.edu Fri Feb 13 17:16:24 1998 Return-Path: Received: from schubert.mit.edu (SCHUBERT.MIT.EDU [18.87.0.16]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id RAA25168 for ; Fri, 13 Feb 1998 17:16:24 -0500 (EST) Received: (from sara(at-sign)localhost) by schubert.mit.edu (8.8.7/8.8.7) id RAA15062; Fri, 13 Feb 1998 17:16:22 -0500 (EST) Date: Fri, 13 Feb 1998 17:16:22 -0500 (EST) Message-Id: <199802132216.RAA15062(at-sign)schubert.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: changes X-URL: http://www-math.mit.edu/~sara There has been a change in the seminar schedule for next week. **The talk scheduled for February 20 by Eva Feitchner will be postponed until March 11. ** February 18: Harald Helfgott Recent developments in enumeration of tilings February 25: Richard Stanley Enumerating solutions to equations in finite groups **Pretalk at 3:30pm in 2-338*** February 27: Alex Burstein Enumeration of words with forbidden patterns ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From kcollins(at-sign)mail.wesleyan.edu Wed Feb 18 11:56:30 1998 Return-Path: Received: from mail.wesleyan.edu (dns.wesleyan.edu [129.133.12.10]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id LAA29533; Wed, 18 Feb 1998 11:24:33 -0500 (EST) Received: from [129.133.30.205] (kcollins1.math.wesleyan.edu [129.133.30.205]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id LAA10467; Wed, 18 Feb 1998 11:16:18 -0500 (EST) X-Sender: kcollins(at-sign)mail.wesleyan.edu Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 18 Feb 1998 11:22:05 -0500 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: third announcement Come to the Twenty-eighth one day conference on Combinatorics and Graph Theory Saturday, February 21, 1998 10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063 Schedule 10:00 Katalin Vesztergombi (Yale University) Properties of Distance-Graphs 11:10 Sheila Sundaram (Danbury, Connecticut) Homology of Graph Complexes and Partitions with Forbidden Block Sizes 12:10 Lunch 2:00 Emily H. Moore (Grinnell College and Mt. Holyoke College) Extending Graph Colorings 3:10 Dana Randall (Georgia Institute of Technology) Two Dimensional Tilings in Two and Three Dimensions The conferences are supported by an NSF grant which allows us to provide a modest transportation allowance to those attendees who are not local. We also gratefully acknowledge support from Smith College and Wesleyan University. Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)math.smith.edu Karen Collins (Wesleyan University), (860) 685-2169, kcollins(at-sign)wesleyan.edu Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)math.smith.edu From sara(at-sign)math.mit.edu Wed Feb 18 12:49:15 1998 Return-Path: Received: from schubert.mit.edu (SCHUBERT.MIT.EDU [18.87.0.16]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id MAA03712 for ; Wed, 18 Feb 1998 12:49:15 -0500 (EST) Received: (from sara(at-sign)localhost) by schubert.mit.edu (8.8.7/8.8.7) id MAA07506; Wed, 18 Feb 1998 12:49:14 -0500 (EST) Date: Wed, 18 Feb 1998 12:49:14 -0500 (EST) Message-Id: <199802181749.MAA07506(at-sign)schubert.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Date: Wednesday, February 18, 1998 Speaker: Harald Helfgott Title: Recent developments in enumeration of tilings Abstract: In this talk I will describe recent work with Ira Gessel on new methods for enumerating tilings of subsets of the Aztec diamond. I will show how these new tools allow us to solve several of Jim Propp's open problems on enumeration of tilings, as well as explain the connection between Eric Kuo's graphical condensation procedure and Dodgson condensation. I will also discuss the use of our methods in the computation of the entries of the inverse Kasteleyn matrix of the Aztec diamond. Upcoming Events: February 20: Eva Feitchner Postponed until March 11. February 25: Richard Stanley (with Pretalk at 3:30) Enumerating solutions to equations in finite groups February 27: Alex Burstein Enumeration of words with forbidden patterns ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From sara(at-sign)math.mit.edu Wed Feb 25 13:49:30 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA09627 for ; Wed, 25 Feb 1998 13:49:30 -0500 (EST) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id NAA12971; Wed, 25 Feb 1998 13:49:29 -0500 (EST) Date: Wed, 25 Feb 1998 13:49:29 -0500 (EST) Message-Id: <199802251849.NAA12971(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 ** Pretalk at 3:30-4:00** Date: Wednesday, February 25, 1998 Speaker: Richard Stanley Title: Enumerating solutions to equations in finite groups Abstract: We will survey such questions, going back to Frobenius and Schur, as: In how many ways can an element w of a finite group G be written as a square or as a commutator? If f(w) is this number, then when is f a character of G? How many n-tuples of elements of G pairwise commute? Special emphasis will be given to the symmetric group S_n. Techniques involve combinatorics, character theory, and symmetric functions. Applications to permutation enumeration will be discussed. A link to everything mentioned in the talk (and more) may be found at http://www-math.mit.edu/~rstan/ec/ec.html. Note: There will be a pretalk for non-experts from 3:30-4:00. Upcoming events: February 27: Alex Burstein Enumeration of words with forbidden patterns March 4: Peter Winkler Phase transitions on a tree March 6: Sara Billey Pattern avoidance and rational smoothness of Schubert varieties March 11: Eva Feichtner On the cohomology algebras of complex subspace arrangements March 13: Takayuki Hibi Nonregular unimodular triangulations of convex polytopes March 18: Alex Suciu Characteristic varieties of real and complex arrangements **Spring Break March 23-27** ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From sara(at-sign)math.mit.edu Wed Mar 4 12:50:23 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id MAA27824 for ; Wed, 4 Mar 1998 12:50:23 -0500 (EST) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id MAA26193; Wed, 4 Mar 1998 12:50:22 -0500 (EST) Date: Wed, 4 Mar 1998 12:50:22 -0500 (EST) Message-Id: <199803041750.MAA26193(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Date: Wednesday, March 4, 1998 Speaker: Peter Winkler (Bell Labs) Title: Phase Transitions On A Tree Abstract: Although it is manifestly different from a cubic lattice in Euclidean space, the Cayley tree (or "Bethe Lattice") offers an opportunity to prove theorems about phase transitions, on account of its connection with branching random walks. Modelling hard-constraint systems by random homomorphisms of graphs, we characterize the constraint graphs which exhibit more than one simple invariant Gibbs measure. Our methods are elementary and no acquaintance with statistical physics will be assumed. Joint work with Graham Brightwell (London School of Economics). Upcoming Events: March 6: Sara Billey **PRETALK at 3:30*** Pattern avoidance and rational smoothness of Schubert varieties March 11: Eva Feichtner On the cohomology algebras of complex subspace arrangements March 13: Takayuki Hibi Nonregular unimodular triangulations of convex polytopes March 18: Alex Suciu Characteristic varieties of real and complex arrangements ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From propp(at-sign)math.mit.edu Thu Mar 5 15:37:27 1998 Return-Path: Received: from pfaff.mit.edu (PFAFF.MIT.EDU [18.87.0.183]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id PAA02907 for ; Thu, 5 Mar 1998 15:37:26 -0500 (EST) Received: (from propp(at-sign)localhost) by pfaff.mit.edu (8.8.7/8.6.9) id PAA01813 for combinatorics(at-sign)math.mit.edu; Thu, 5 Mar 1998 15:37:26 -0500 (EST) Date: Thu, 5 Mar 1998 15:37:26 -0500 (EST) From: Jim Propp Message-Id: <199803052037.PAA01813(at-sign)pfaff.mit.edu> To: combinatorics(at-sign)math.mit.edu Subject: Winkler preprints Copies of Peter Winkler's three articles Graph homomorphisms and phase transitions (with Graham Brightwell) Gibbs measures and dismantleable graphs (with Graham Brightwell) Nonmonotonic behavior in hard-core and Widom-Rowlinson models (with Graham Brightwell and Olle Haggstrom) are now available on the ledge in the public area in 2-363. Jim Propp From sara(at-sign)math.mit.edu Fri Mar 6 13:36:08 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA25484 for ; Fri, 6 Mar 1998 13:36:08 -0500 (EST) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id NAA04637; Fri, 6 Mar 1998 13:36:08 -0500 (EST) Date: Fri, 6 Mar 1998 13:36:08 -0500 (EST) Message-Id: <199803061836.NAA04637(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 ***Pretalk at 3:30 in 2-338*** Date: Friday, March 6, 1998 Speaker: Sara Billey (MIT) Title: Pattern avoidance and rational smoothness of Schubert varieties Note: This talk will be a joint seminar with the algebraic geometers. Abstract: Let w be an element of the Weyl group S_n, and let X_w be the Schubert variety associated to w in the flag manifold SL_{n}(C)/B. Lakshmibai and Sandhya showed that X_w is smooth if and only if w avoids the patterns 4231 and 3412. Using two tests for rational smoothness due to Carrell, we show that rational smoothness of X_w is characterized by pattern avoidance for types B and C as well. A key step in the proof of this result is a sequence of rules for factoring the Poincare polynomials for the cohomology ring of X_w generalizing the recent work of Gasharov. Preprints Available: http://www-math.mit.edu/~sara/ Upcoming Events: March 11: Eva Feichtner On the cohomology algebras of complex subspace arrangements March 13: Takayuki Hibi Nonregular unimodular triangulations of convex polytopes March 18: Alex Suciu Characteristic varieties of real and complex arrangements ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From sara(at-sign)math.mit.edu Wed Mar 11 13:48:45 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA29333 for ; Wed, 11 Mar 1998 13:48:45 -0500 (EST) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id NAA13681; Wed, 11 Mar 1998 13:48:44 -0500 (EST) Date: Wed, 11 Mar 1998 13:48:44 -0500 (EST) Message-Id: <199803111848.NAA13681(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 ******Pretalk at 3:30 in 2-338****** Date: Wednesday, March 11, 1998 Speaker: Eva Feichtner (MIT) Title: On the cohomology algebras of complex subspace arrangements Abstract: We show that for a complex subspace arrangement with geometric intersection lattice the integral cohomology algebra of the complement is completely determined by the combinatorial data of the arrangement. Our approach relies on a combinatorial encoding of the arrangement's topology into order complexes of posets. We explicitly describe generating cohomology classes on simplicial models for the complement of the arrangement and derive a presentation of its cohomology algebra in the spirit of the combinatorial description of cohomology algebras of complex hyperplane arrangements by Orlik and Solomon. This is joint work with Gunter M. Ziegler. Upcoming Events: March 13: Takayuki Hibi Nonregular unimodular triangulations of convex polytopes March 18: Alex Suciu Characteristic varieties of real and complex arrangements ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From kcollins(at-sign)mail.wesleyan.edu Fri Mar 13 12:17:13 1998 Return-Path: Received: from mail.wesleyan.edu (dns.wesleyan.edu [129.133.12.10]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id MAA24002; Fri, 13 Mar 1998 12:17:13 -0500 (EST) Received: from [129.133.30.205] (kcollins1.math.wesleyan.edu [129.133.30.205]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id MAA25331; Fri, 13 Mar 1998 12:16:20 -0500 (EST) X-Sender: kcollins(at-sign)mail.wesleyan.edu Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Fri, 13 Mar 1998 12:20:43 -0500 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: April 4th CoNE meeting Come to the Twenty-ninth one day conference on Combinatorics and Graph Theory Saturday, April 4, 1998 10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063 Schedule 10:00 Van Vu (Yale University) Witness on the Upper Bound of Chromatic (Choice) Number of Random Graphs 11:10 Richard Stanley (MIT) Spanning Trees and a Conjecture of Kontsevich 12:10 Lunch 2:00 Eckhard Steffen (Princeton University) Snarks 3:10 Therese Biedl (McGill University) Efficient Algorithms For Petersen's Theorem The next meeting is scheduled for May 2nd. The conferences are supported by an NSF grant which allows us to provide a modest transportation allowance to those attendees who are not local. We also gratefully acknowledge support from Smith College and Wesleyan University. Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)math.smith.edu Karen Collins (Wesleyan University), (860) 685-2169, kcollins(at-sign)wesleyan.edu Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)math.smith.edu From sara(at-sign)math.mit.edu Fri Mar 13 13:01:55 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA25282 for ; Fri, 13 Mar 1998 13:01:54 -0500 (EST) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id NAA17185; Fri, 13 Mar 1998 13:01:54 -0500 (EST) Date: Fri, 13 Mar 1998 13:01:54 -0500 (EST) Message-Id: <199803131801.NAA17185(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 ******Pretalk at 3:30 in 2-338****** Date: Friday, March 13, 1998 Speaker: Takayuki Hibi(Osaka University) Title: Nonregular unimodular triangulations of convex polytopes Abstract: It was unknown, for a long time, if there exists a lattice polytope which possesses a unimodular triangulation and none of whose unimodular triangulations is regular. The purpose of my talk is to present a family ${\cal F}$ of $(0,1)$-polytopes such that each polytope belonging to ${\cal F}$ has a unimodular triangulation, but has no regular unimodular triangulation. This is a joint work with Hidefumi Ohsugi. Upcoming Events: March 18: Alex Suciu Characteristic varieties of real and complex arrangements April's schedule will be available soon. The ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From gcrota(at-sign)earthlink.net Sun Mar 15 16:43:51 1998 Return-Path: Received: from sweden.it.earthlink.net (sweden-c.it.earthlink.net [204.250.46.50]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id QAA00759 for ; Sun, 15 Mar 1998 16:43:46 -0500 (EST) Received: from [153.35.78.59] (1Cust59.max18.boston.ma.ms.uu.net [153.35.78.59]) by sweden.it.earthlink.net (8.8.7/8.8.5) with ESMTP id NAA22609; Sun, 15 Mar 1998 13:42:39 -0800 (PST) Date: Sun, 15 Mar 1998 13:42:39 -0800 (PST) Message-Id: Mime-Version: 1.0 Content-Type: multipart/mixed; boundary="============_-1322142191==_============" To: bjorner(at-sign)fredholm.math.kth.se, combinatorics(at-sign)math.mit.edu From: Gian-Carlo Rota Subject: Colloquium Lectures --============_-1322142191==_============ Content-Type: text/plain; charset="us-ascii" Enclosed is a dvi files of the text of the Colloquium Lectures I delivered at the Annual Meeting of the American Mathematical Society in January of this year. I will appreciate any comments of yours before I draft the final text. 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'BfeLH$%bmbNK)LbD!!`!!!!+!!!!"Q0YFhNa-2-UI(YC"`!)!!!!#!!!!!4MEA) imbZj3(at-sign)'S!!B!!!!'!!!!"'0YFMEc,,IKCk-!$!!!!!`!!!!'BfeYD6%bmbhA!4F b!!J!!!!)!!!!"(at-sign)0YE(at-sign)Nimbi3cEl1!!B!!!!'!!!!"(at-sign)0YE(at-sign)Nfmbqq5mJ,!!J!!!! )!!!!"(at-sign)0YFhNimc"aS58,!!B!!!!'!!!!"(at-sign)0YFhNfmc%bad$*!!J!!!!)!!!!"(at-sign)0 YBRJimcQEZiK!!!`!!!!-!!!!"Q0YG'Na-[N!!R2h!YrIhprIhpp+q3!!: --============_-1322142191==_============-- From gcrota(at-sign)earthlink.net Sun Mar 15 22:46:17 1998 Return-Path: Received: from sweden.it.earthlink.net (sweden-c.it.earthlink.net [204.250.46.50]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id WAA05179 for ; Sun, 15 Mar 1998 22:46:12 -0500 (EST) Received: from [153.35.78.66] (1Cust66.max18.boston.ma.ms.uu.net [153.35.78.66]) by sweden.it.earthlink.net (8.8.7/8.8.5) with ESMTP id TAA12864 for ; Sun, 15 Mar 1998 19:45:35 -0800 (PST) Date: Sun, 15 Mar 1998 19:45:35 -0800 (PST) Message-Id: Mime-Version: 1.0 Content-Type: multipart/mixed; boundary="============_-1322120415==_============" To: combinatorics(at-sign)math.mit.edu From: Gian-Carlo Rota Subject: Colloquium Lectures --============_-1322120415==_============ Content-Type: text/plain; charset="us-ascii" I am sending you a new version, since I have received complaints that the one sent out earlier cannot be printed except with a Macintosh. G.-C. R. --============_-1322120415==_============ Content-Type: text/plain; name="colloquium_3=14=98_2"; charset="us-ascii" Content-Disposition: attachment; filename="colloquium_3=14=98_2" \documentstyle[12pt]{article} \def\bphiz#1{\overline\phi_{\hat0}^{#1}} \def\bPhi{\overline\Phi} \def\phiz#1{\phi_{\hat0}^{#1}} \def\sigalg#1{\Sigma^{\otimes{#1}}} \def\sigalgpi#1{\Sigma^{\otimes{#1}}_\pi} \def\finalg#1{\Sigma^{\otimes{#1}}_{\rm fin}} \def\finalgpi#1{\Sigma^{\otimes{#1}}_{{\rm fin},\pi}} \def\ltimes{{\rm X}} \def\D{{\cal D}} \def\F{{\cal F}} \def\A{{\cal A}} \def\B{{\cal B}} \def\E{{\cal E}} \def\C{{\cal C}} \def\G{{\cal G}} \def\R{{\bf R}} \def\S{{\bf S}} \def\Inpi{{I^{[n]}_\pi}} \def\bs{\backslash} \def\zpione{{\hat 0 \le \pi \le \hat 1}} \def\zpilone{{\hat 0 \le \pi < \hat 1}} \def\halmos{{Q.E.D.}} % This is temporary \def\St{{\rm St}} \newtheorem{definition}{Definition} \newtheorem{lemma}{Lemma} \newtheorem{theorem}{Theorem} \begin{document} \setlength{\textwidth}{6in} \setlength{\textheight}{8in} \parindent=0pt \tolerance=6000 \title{The American Mathematical Society Colloquium Lectures, 1998} \author{Gian-Carlo Rota} \date{Baltimore, January 7, 8, and 9, 1998} \maketitle \newpage \begin{center} \bigskip {\bf Contents:} \end{center} \begin{align} 1. Introduction to geometric probability. \bigskip 2. Invariant theory, old and new. \bigskip 3. Combinatorial snapshots: \bigskip \begin{itemize} \item First snapshot: an example of profinite combinatorics. \item Second snapshot: the cyclic derivative. \item Third snapshot: logarithms and the binomial theorem. \end{itemize} \end{align} \newpage \newpage \begin{center} {\bf INTRODUCTION TO GEOMETRIC PROBABILITY}\\ being\\ The first of three colloquium Lectures\\ delivered at the Annual Meeting of the American Mathematical Society\\ Baltimore, January 7, 1997\\ Gian-Carlo Rota\\ Department of Mathematics\\ MIT\\ Cambridge MA 02139-4307\\ \end{center} \bigskip \bigskip I am very happy to be here before you as the Colloquium Lecturer for this year, and I feel deeply honored to be given this great opportunity to share with you some of the mathematics we love. \bigskip When I received from Bob Fossum the invitation to be the Colloquium Lecturer for this year, I rushed to the library to read in an old issue of the Notices the list of all previous Colloquium Lecturers, going all the way back to James Pierpont in 1896. It is a list of distinguished mathematicians, and I wondered how my name could ever belong in such lofty company. My immediate temptation was to decline the invitation outright; but Bob Fossum assured me that no one in the history of the Society has ever declined the invitation to be the Colloquium Lecturer. So I went back to the list of previous colloquium speakers, in search for a justification of my presence in that list. As often happens in such situations, I soon enough found such a justification. I computed the average age of colloquium lecturers and discovered that this average is somewhat lower than my age, as a matter of fact my age exceeds by approximately one standard deviation the mean age of previous speakers. As I came to this realization, I began to fantasize on the probable topics that my younger predecessors might have chosen. I imagined a brilliant young mathematician, eager to establish himself as a leader in his field, delivering one single dazzling proof beginning with the first colloquium lecture and lasting all the way to the end of the third. Or else, some middle aged mathematician, anxious to have his latest theory accepted by the mathematical world, delivering to a thrilled audience a three-hour condensation of material that would normally take an entire term in an advanced graduate course. How could I, a mathematician one standard deviation older, ever hope to match such enviable feats? \bigskip These fantasies came to an abrupt end when Bob Fossum informed me in no uncertain terms that the Council of the Society had decided that the three colloquium lectures must deal with three independent and unrelated topics, thereby allowing any member of the audience to skip one or more lectures, without missing anything. Bob Fossum's command deprived me of all possible role models among previous colloquium lecturers. In a state of temporary panic, I again scanned the list of previous colloquium speakers, this time looking for names of mathematicians who had not been chosen for this honor. Sure enough, one name was conspicuously missing: that of Hermann Weyl. \bigskip I hope you will forgive me if I digress with some personal reminiscences. \bigskip In the fall of 1950 I enrolled as a freshman at Princeton, having graduated a few months before from the American High School of Quito, Ecuador. The principal of the American High School of Quito was a Princeton graduate, and he steered me towards Princeton University. \bigskip In November 1950 I listened to my first mathematics lectures. These were the three Vanuxem lectures, delivered by Hermann Weyl and bearing the generic title "Symmetry". These lectures were an unforgettable experience. The lectures took place in the old chemistry auditorium, packed with an expectant public. As I shamelessly sat in the first row trying to guess which of the other persons sitting in the same row was to be the speaker, a hush fell upon the audience: Einstein was entering the lecture room. To my disappointment, he sat somewhere in the middle of the auditorium. \bigskip The first lecture began with an impressive and lengthy quotation in Greek, which no one in the audience understood except Luther Pfahler Eisenhart. This brilliant start was followed by a display of slides portraying charming women wearing the long brimmed hats fashionable at the time, and later by more slides showing the Alhambra and the Pentagon. Not a word of mathematics. The audience was left wondering where such a sparkling display of "Kultur" was leading up to. Not much more mathematics was mentioned in the second lecture, when more slides were shown of physics experiments, for which the lecturer provided a learned oral commentary. Only in the last lecture did some group theory make a modest appearance. By that time the audience, which had not dwindled, was enthralled with the subject, and did not mind the fact that the speaker had said very little about mathematics, actually he had said very little about anything at all. What is more remarkable, the audience seemed to be thankful to the speaker for making the contents of the three lectures independent of one another, thereby minimizing all memory requirements. I hazard to guess that the success of Hermann Weyl's lectures may be in part attributed to the speaker's astute foresight in making his lectures self contained, independent and lightweight. As I recall this distant episode, I realize that Bob Fossum's injunction about the independence of the present colloquium lectures is a wise one, all the more so when the speaker is not Hermann Weyl. \bigskip You may wonder why I cited my age as an aid to delivering these colloquium lectures. What difference does one standard deviation make? I think it makes some difference. It is a relief, both to you and to me, to know right at the start that the speaker does not feel the need to impress you by stating the results of his latest research. Nor do you or I suffer from any lack of exposure to the latest fashions in mathematics; we'll hear enough about them in other lectures scheduled to be delivered at this meeting. We can therefore afford to spend these three hours on leisurely discussion of some mathematics that may matter to both you and me. \bigskip We will cover in these lectures a few items that are not widely known, that should be better known, and that I vouch can be understood by anyone with a B.A. in mathematics. I solemnly promise that I will not state any big theorems, I will not subject you to any ingenious arguments, and that I will not announce any revolutionary developments. \bigskip The title of this lecture is "Geometric Probability". A definition of geometric probability might run as follows: geometric probability is the study of invariant measures. Like all definitions, this does not tell us anything until we are shown some typical examples, and these examples are the content of this lecture. \bigskip About one hundred years ago, the properties that underlie such notions as length, area, volume, as well as the probability of events were abstracted under the banner of the word "measure". Let us review the definition of measure, since we will be using this definition in an unusual way. \bigskip A measure $\mu$ is a function defined on a family of subsets of a set $S$, which takes real values, not necessarily positive. The family of sets on which a measure is defined is closed under unions and intersections, and contains the empty set. \bigskip A measure is characterized by two simple axioms. Let us take a minute to review these axioms. \bigskip Axiom 1. $$ \mu ( \emptyset ) = 0,$$ where $\emptyset $ is the empty set. \bigskip Axiom 2. If $A$ and $B$ are two measurable sets, then $$\mu(A\cup B) = \mu (A) +\mu(B) - \mu (A\cap B) .$$ \bigskip The meaning of this second axiom is clear. The axiom states that measure is additive. In particular, if we have two disjoint sets $A$ and $B$, then $$\mu(A\cup B) = \mu (A) + \mu (B) .$$ \bigskip More generally, for any finite family $F$ whose members are sets, and for which any two members are disjoint, we have : $$\mu( \bigcup_{A\in F} A ) =\sum_{A\in F}\mu (A).$$ We most emphatically do not assume that a measure is countably additive. \bigskip The best known example of a measure is the volume $\mu_n(A)$ of a solid $A$ in ordinary $n$-dimensional Euclidean space. The volume $\mu_n (A)$ of a solid $A$ satisfies axioms 1 and 2 above, but axioms 1 and 2 do not characterize volume among all possible measures. What additional axioms must we add to the definition of a measure, in order to characterize volume? It is possible to characterize volume among all measures by adding to axioms 1 and 2 two additional intuitive axioms, namely, the following: \bigskip Axiom 3. \bigskip The volume of a set $A$ is independent of the position of $A$. If a set $A$ in $n$-dimensional Euclidean space can be rigidly moved onto a set $B$, then $A$ and $B$ have the same volume. \bigskip In other words, volume is invariant under the group of Euclidean motions. Lastly, we must prescribe a normalization, as physicists say. This is done by taking a parallelotope $P$ with orthogonal sides of lengths $ x_1, x_2, \dots , x_n $, and setting \bigskip Axiom 4. $$ \mu_n (P) = x_1x_2 \dots x_n.$$ \bigskip These axioms, together with suitable continuity conditions, uniquely determine the volume of solids in Euclidean $n$- space. For example, starting from these four axioms, by a limiting process such as one finds in an advanced calculus textbook, one establishes the fact that the volume of a ball $S_r$ of radius $r$ in $n$-dimensional space is given by the following formulas: $$\mu_n(S_r) = \frac{ \pi^{n/2} r^n}{(n/2)!} $$ if the dimension $n$ is even and $$\mu_n (S_r) = \frac{2^n \pi^{(n-1)/2}((n-1)/2)! r^n}{n!} $$ if the dimension $n$ is odd. \bigskip It is still widely believed that volume is the only invariant measure in Euclidean $n$-space. But in point of fact there are other invariant measures, defined on all reasonable subsets of Euclidean $n$-space, which have a notable geometric significance. Our objective is to describe all such invariant measures. \bigskip What happens if we keep the first three axioms, but tamper with the fourth axiom, the normalization axiom ? Will we get something interesting, or will we get nothing new? To answer this question, we will appeal to the basic tools of combinatorial mathematics. \bigskip The basic tools of combinatorial mathematics are the elementary symmetric functions, to wit, the following polynomials in $n$ variables: \begin{align} $$e_1(x_1,x_2,...,x_n) = x_1 + x_2 + ... +x_n.,$$ $$e_2(x_1,x_2,...,x_n) = x_1x_2+x_1x_3+...+x_{n-1}x_n,$$ $$ \dots $$ $$e_{n-1}(x_1, x_2, \cdots, x_n) = x_2x_3 \cdots x_n + x_1x_3x_4 \cdots x_n+...+ x_1x_2 \cdots x_{n-1},$$ $$ e_n(x_1, x_2, \cdots , x_n) = x_1 x_2\cdots x_n .$$ \end{align} Observe an interesting coincidence. The last of these $n$ symmetric functions is also the formula for the volume of a parallelotope. Axiom 4 can be rewritten as Axiom 4. $$\mu_n (P) = e_n(x_1,x_2,\cdots , x_n).$$ \bigskip Let us try an experiment, and replace the $n$-th symmetric function by the $n-1$-st symmetric function. Let us first take $n = 3$, that is, three-dimensional space, so that we can better visualize what is going on. Let us see whether we can define a measure on subsets of $3$-dimensional space by keeping three of the above axioms, but by replacing the normalization Axiom 4 by using another symmetric function instead of the symmetric function $e_3(x_1,x_2,x_3)$ which gives the volume. Let us first replace the symmetric function $e_3$ by the symmetric function $e_2$, thereby changing Axiom 4 to \bigskip Axiom 4': $$\mu_2(P) = x_1x_2 + x_1x_3 + x_2x_3 . $$ Does this axiom define a measure? Of course it does. The right hand side is the formula for the surface area of the parallelotope $P$, divided by $2$. Again we will find in any advanced calculus textbook the explanation of the fact that axioms 1, 2, 3, and 4', together with some continuity considerations, completely determine an invariant measure which is the surface area of solids in ordinary space. For example, the following well known formula for the surface area of a ball $S_r$ of radius $r$ in $3$ dimensions is obtained from these axioms: $$\mu_2(S_r)= 4\pi r^2 .$$ Let us take the next step. \bigskip Emboldened by our success with two symmetric functions, we now replace axiom 4 by yet another axiom, using another symmetric function. Let us set \bigskip Axiom 4". $$ \mu_1(P) = e_1(x_1, x_2, x_3) = x_1 + x_2 + x_3 $$ The new measure $\mu_1$ will satisfy Axioms 1, 2, and 3, and in addition it will satisfy axiom 4". The symmetric function of degree one plays the role that in the previous two examples was played by the other two symmetric functions. \bigskip But wait a minute: is this definition consistent? \bigskip To realize that the definition of the new measure $\mu_1$ is consistent, that is, that $\mu_1$ as defined by axioms 1, 2, 3, and 4" really exists and is not a dream of reason, look at two parallelotopes $P_1$ and $P_2$ that have a face in common. The first parallelotope has sides equal to $x_1, x_2, x_3$ , and the second parallelotope has sides equal to $x_1, x_2, y $. The two parallelotopes have a common face with sides equal to $x_1, x_2$. The measure $\mu_1(P_1\cup P_2)$ of the parallelotope $P_1\cup P_2 $ can be computed in two ways: using the left side of axiom 2, or using the right side, and the two computations had better yield the same answer, in symbols: $$\mu_1(P_1 \cup P_2) = \mu_1(P_1) + \mu_1(P_2) - \mu_1(P_1 \cap P_2).$$ Let us check this. \bigskip The left side is computed by observing that the parallelotope $P_1\cup P_2$ has sides equal to $x_1$,$ x_2$ and $x_3 + y$. Therefore, Axiom 4" tells us that $$\mu_1(P_1 \cup P_2) = x_1 + x_2 + x_3 + y.$$ Now let us compute the right side. We have $$\mu_1(P_1) = x_1+x_2+x_3$$ $$\mu_1(P_2) = x_1 + x_2 + y$$ $$\mu_1(P_1 \cap P_2) = x_1 + x_2 ,$$ again by Axiom 4" applied to $P = P_1 \cap P_2$, since one side equals zero when the parallelotope is a flat, that is, a rectangle. Therefore, the right side of Axiom 2 equals $$\mu_1(P_1) + \mu_1(P_2) - \mu_1(P_1 \cap P_2 )= x_1+x_2+x_3 + x_1+x_2+ y - (x_1+x_2) = $$ $$ x_1+x_2+x_3+y ,$$ and the two sides of our equations agree, thereby convincing us that the definition may well be consistent. \bigskip The preceding argument is convincing, even though it proves nothing. \bigskip Actually, the definition of $\mu_1(P) $ for a parallelotope $P$ has a simple geometric interpretation. When multiplied by 4, it equals the perimeter of the parallelotope $P$, that is, the sum of the lengths of all the edges of the parallelotope $P$. Just as happens for volume and area, it can be shown by continuity considerations that the measure $\mu_1$ can be extended to all reasonable solids in ordinary space, for example, to all convex sets and to all polyhedra, convex or nonconvex. \bigskip But, one may object, $\mu_1(P)$ makes sense for a parallelotope $P$, because a parallelotope has a well defined perimeter. What if $A$ is a solid that does not have a well defined perimeter, a sphere for example? The definition of the measure $\mu_1(A) $ for solids $A$ that may not have a well defined perimeter flies in the face of common sense. \bigskip Einstein wrote: "Common sense is the residue of those prejudices that were instilled into us before the age of seventeen". Common sense must constantly readjust to reality. \bigskip The new measure $\mu_1$ that we obtain in this way is called the mean width, a misnomer that has been kept for historical reasons. The mean width of a solid in space is completely characterized by axioms 1, 2, 3, and 4". In particular, it is invariant, that is, it does not depend on position. For example, the formula for the mean width of a sphere of radius $r$ is computed to be $$\mu_1(S_r) = 4r. $$ Thus we see that in three dimensions each of the three elementary symmetric functions of three variables leads to an invariant measure that enjoys equal rights with volume. The first two of these measures are well known, namely, volume and area. The third, the mean width, is at present almost totally unknown. I know of no person who has an intuitive feeling for the mean width, similar to the intuitive feeling we have for volume and area. \bigskip Let us conjecture a possible application of the mean width. A potato grower knows that a potato's volume is important, because it determines the nutritional content of the potato. The potato grower also knows that the surface area of a potato is important, because it is rumored that the vitamins in a potato are concentrated in the skin. We may conjecture that as soon as the potato grower will become aware of the mean width, he or she will find a nutritional interpretation of the mean width of a potato. I am indebted to Steve Schanuel for this example. \bigskip A similar kind of reasoning works in $n$ dimensions. We discover $n$ different invariant measures, each of them well defined on all polyhedra and on all finite unions of compact convex sets. Each of the $n$ elementary symmetric functions of $n$ variables leads to the definition of a new invariant measure which is a different generalization of the notion of volume. These $n$ measures are called the intrinsic volumes. The intrinsic volumes are first defined on an orthogonal polytope $P$ whose sides equal $x_1,x_2,\dots , x_n$ by setting $$\mu_k(P) = e_k(x_1,x_2,\dots ,x_n),$$ where $ e_k(x_1,x_2,\dots ,x_n) $ is the $k$-th elementary symmetric function. Here, the subscript $k$ ranges from $1$ to $n$. \bigskip One then proceeds to extend the definition of the intrinsic volumes to more general sets, by a technique which we will shortly see. \bigskip The intrinsic volumes are independent of each other, except for certain inequalities they satisfy. Mathematicians are presently working on determining these as yet unknown inequalities among the intrinsic volumes. These inequalities generalize the classical isoperimetric inequality that relates volume to area. At present, we know very little about the intrinsic volumes; they have not been around for long and very little research has been done on them. We do not even know the formula for the intrinsic volumes of an $n$-simplex. \bigskip Now you are thinking: this is all fine and dandy, but how is the extension of the intrinsic volumes from parallelotopes to more general sets carried out? And besides, isn't there any intuitive interpretation we can give the intrinsic volumes? \bigskip We will answer both these questions simultaneously. Let us go back to three-dimensional space. You all know that the set of all straight lines in space - not necessarily through the origin - forms a nice algebraic variety, called the Grassmannian. The group of all Euclidean rigid motions acts on the Grassmannian, and there is an invariant measure on the Grassmannian under the action of the group of Euclidean motions. This invariant measure is unique except for a constant factor. A similar statement may be made about the set of all planes, and more generally for the set of all linear varieties of dimension $k$ in Euclidean space of dimension $n$. Remember that these linear varieties need not pass through the origin. \bigskip In the practice of mathematics, computation with invariant measures on Grassmanians is rare; most mathematicians would be hard put even to recall an explicit formula for the invariant measures on Grassmanians. Let us take a few minutes to get a feeling for the invariant measure on the set of all straight lines in three-space. As is customary, we begin by giving this measure a name: let us call it $\lambda^3_1$; the upper index $3$ stands for three-dimensional space, and the lower index stands for the dimension of a line, namely, one. To repeat, we use the notation $\lambda^3_1$ to denote the invariant measure on the set of all straight lines in three-space. \bigskip Consider a rectangle $R$ placed anywhere in space, and consider the set of all straight lines that meet the rectangle $R$. Can we compute the measure of this set of lines without knowing the formula for the invariant measure on the Grassmannian of all lines in three-space? Of course we can. A straight line meets the rectangle $R$ either at a point or not at all; therefore, the value of the measure of the set of all lines meeting $R$ depends only on the area $\mu_2(R)$ of the rectangle $R$. If we take another rectangle $R'$ whose area is double the area of $R$, then the measure of the set of all lines meeting $R'$ is double the measure of the set of all lines meeting $R$. Proceeding along these lines, we get to Cauchy's functional equation, and we infer that the measure of the set of all straight lines meeting a rectangle $R$ equals a constant times the area $\mu_2(R)$. Since we are at liberty to choose a normalization of the measure, let us agree to set this constant equal to one. \bigskip But instead of working with a rectangle we could have worked with any planar figure $C$ whatsoever, placed in an arbitrary position in space. The measure of the set of lines meeting $C$ equals the area $\mu_2(C)$, by the same reasoning. We stress the assumption that $C$ must lie in a plane. To conclude: even without knowing the formula for the invariant measure $\lambda^3_1$, we can nevertheless compute the value of such a measure on certain sets of lines. \bigskip Let us now take a more sophisticated set of straight lines. We take a set $D$ in three-space that is the union of disjoint sets $C_1, C_2,\cdots, C_n$, where each of the $C_i$ is contained in a different plane, and we ask for the measure of the set of all straight lines meeting $D$. Such a computation can be carried out, but it is a combinatorial nightmare; so much so, that we are forced to do what mathematicians do when confronted with combinatorial nightmares: they change the problem ever so slightly. In this case we take a hint from the way probabilists work. Let $X_D(\omega)$ equal the number of times the straight line $\omega$ meets the set $D$. Instead of computing a measure, let us compute the integral $$ \int X_D(\omega) d\lambda^3_1(\omega) ,$$ where $\omega$ ranges over the Grassmannian, that is, over the set of all straight lines in space. We will see that we can compute this integral without knowing the measure $\lambda^3_1$ on the Grassmannian. Since $$ D = \bigcup_{i=1}^n C_i ,$$ and since the $C_i$ are disjoint, we have $$ \int X_D(\omega) d\lambda^3_1(\omega) = \sum_{i=1}^n \int X_{C_i}(\omega) d\lambda^3_1(\omega). $$ But we have chosen each of the sets $C_i$ to lie in a plane, so that a straight line meets $C_i$ either once or not at all. It follows that $$ \int X_{C_i}(\omega) d\lambda^3_1(\omega) = \mu_2(C_i) $$ and therefore $$ \int X_D(\omega) d\lambda^3_1(\omega) = \sum_{i=1}^n \mu_2(C_i) . $$ What is this identity telling us? The right hand side equals the area of the surface $D$. Nothing stops us from passing to the limit, and making the following assertion. Let $E$ be "any" surface in space, and let $X_E(\omega) $ be the number of times the straight line $\omega$ meets the surface $E$. Then the integral $$ \int X_E(\omega) d\lambda^3_1(\omega) $$ ranging over all straight lines $\omega$, equals the surface area of $E$. In probabilistic language: the average number of times a randomly chosen straight line meets the surface $E$ equals the surface area of $E$. \bigskip Let us now retrace our steps, and repeat the same reasoning taking the set of all planes in space, instead of the set of all straight lines. The invariant measure on this Grassmannian is denoted by $\lambda_2^3$, where again the upper index stands for three dimensional space, and the lower index for the dimension of a plane. Since a plane meets a straight line segment either at a point or not at all, the same argument shows that the measure of the set of all planes that meet a line segment $L$ equals $\mu_1(L)$, namely, the length of the segment $L$; more generally, if $F$ is any curve "whatsoever" in space, and if $X_F(\omega)$ equals the number of times the plane $\omega$ meets the curve $F$, then repeating the argument we used for straight lines we infer that the integral $$ \int X_F(\omega) d\lambda^3_2(\omega) $$ equals the length of the curve $F$. The variable of integration $\omega$ now ranges over planes, not over straight lines. Here again we compute an integral without knowing the measure. \bigskip We are now very close to getting an intuitive interpretation of the mean width. Recall the parallelotope $P$ with sides equal to $x_1,x_2,x_3$. To measure the planes meeting the parallelotope $P$, we first consider a family of {\em parallel} planes, all sharing the same fixed unit normal $u$. In other words, consider the set of all planes parallel to the plane $u^{\perp}$. Without loss of generality, place the parallelotope in space so that one of the vertices of $P$ is at the origin, and such that the vector $u$ lie in the octant of space opposite to the parallelotope $P$. (We can do this generically.) Denote the edges of $P$ that meet the origin by $x_1,x_2,x_3$. Given the fixed unit vector $u$, and its family of normal planes, let us take the curve $F$ to be a path along the edges (line segments) $$[0,x_1], \; [x_1,(x_1+x_2)], \; \hbox{ and } \; [(x_1+x_2),(x_1+x_2+x_3)],$$ in that order. A plane parallel to $u^{\perp}$ meets the parallelotope $P$ if and only if it meets the curve $F$ on the parallelotope at exactly one point. Therefore, the measure of the set of all planes parallel to $u^{\perp}$ that meet the parallelotope $P$ is proportional to the length of the curve $F$. Averaging over all unit vectors $u$ (and hence, over all families of parallel planes), we conclude that the measure of the set of all planes meeting a parallelotope equals the mean width of the parallelotope, except for a constant factor which we will again set to be one. \bigskip In view of this realization, we can immediately see how to define the mean width of any closed convex set: it equals the measure of the set of all planes that meet the convex set. Thus, we have shown that the mean width may be extended to all closed convex sets in space. \bigskip We are now in a position to give a probabilistic interpretation of the mean width of a convex set. \bigskip Take two compact convex sets $A$ and $B$ in three dimensional Euclidean space, and suppose that $A$ is contained in $B$. Let us begin by belaboring the obvious. Suppose that we take a point at random belonging to the larger set $B$. What is the probability that the point shall belong to the smaller set $A$? The answer is clear: such a probability equals the ratio of the volume of $A$ by the volume of $B$. \bigskip Instead of choosing a point at random, let us choose a straight line at random in space. Assuming that such a straight line meets the larger set $B$, what is the probability that such a straight line will also meet the smaller set $A$? We have already computed the answer to this question, albeit implicitly. Such a probability equals the surface area of the set $A$, divided by the surface area of the set $B$. \bigskip You can tell what is coming next. We now take a random plane in space. Assuming that the plane meets the larger set $B$, what is the probability that it will also meet the smaller set $A$? The answer is the following: such a probability equals the mean width of $A$, divided by the mean width of $B$. \bigskip In Euclidean $n$-space, we obtain by much the same reasoning interpretations of the intrinsic volume $\mu_k(C)$ of a compact convex set $C$ as the Grassmannian measure of the set of all linear varieties of dimension $n-k$ that meet the convex set $C$, and a similar probabilistic interpretation holds. \bigskip What comes next? There are at least two questions still open. First, are there any other invariant measures besides the intrinsic volumes, and second, how can the definition of the intrinsic volumes be extended to more general subsets of $n$-space than convex sets. The answers to both these questions are closely related. \bigskip The answer to the first question is negative. We are missing one measure, and to discover it, we will engage for a minute in the kind of mathematical reasoning that physicists find unbearably pedantic, just to show physicists that such reasoning does pay off. \bigskip Let us ask ourselves the question: what is the value of the symmetric function of order zero of a set of $n$ variables $x_1,x_2,\dots , x_n$, say $e_0(x_1,x_2,\dots , x_n)$? I will give you the answer, and will leave it to you to justify this answer after the lecture is over. The answer is the following: $e_0= 1$ if $n >0$, that is, if the set of variables $x_1,x_2,\dots , x_n$ is non empty, and $e_0 = 0 $ if the set of variables is empty. \bigskip We are led to believe that there may exist an invariant measure in $n$-space associated with the symmetric function of order zero. We set $$\mu_0(C) = 1 $$ if $C$ is any non empty compact convex set, and of course $\mu_0(\emptyset ) = 0$. Does such a measure exist? It does indeed exist, and the fact that it exists is, in my opinion, one of the most remarkable discoveries ever made in mathematics. \bigskip We will prove that such a measure is well defined on any set which is a finite union of compact convex sets. We do this by employing a classical device borrowed from functional analysis: instead of defining a measure, we define a linear functional on all simple functions, that is, on all real functions $f(\omega)$ defined for $\omega \in R^n $ which are linear combinations of indicator functions of compact convex sets. Let us first begin with the case $n=1$, that is, let $\omega$ range over points on the line. Define a linear functional $\chi_1$ on simple functions as follows: $$ \chi_1(f) = \sum (f(\omega) - f(\omega+)) ,$$ where the sum ranges over all real numbers $\omega$. The meaning of the plus sign is best gleaned from an example. Let $f$ be the indicator function of the closed segment $[a,b]$. Then $f(\omega) - f(\omega+) = 0$ for all $\omega $ except $\omega = b $, because we have $f(b) = 1$ but $f(b+) = 0$. Thus, we see that $\chi_1(f) = 1 $ if $f$ is the indicator function of an interval $[a,b]$. \bigskip Now let us go over to $n$ dimensions, proceeding by induction. Do not worry, this won't take long. Take a straight line $L$ and for every point $\omega$ in $L$ let $H_{\omega} $ be the hyperplane through the point $\omega $ perpendicular to the line $L $. If $f$ is a simple function defined in $n$ space, and if $\omega$ is a point on the straight line $L$, let $f_{\omega}$ be the restriction of $f$ to the hyperplane $H_{\omega} $. Define a linear functional $\chi_n $ as follows: $$ \chi_n(f) = \sum \chi_{n-1}(f_{\omega}) - \chi_{n-1}(f_{\omega +}) ,$$ where the sum ranges over all points $\omega$ on the line $L$. There is only a finite set of $\omega's $ for which the summand is nonzero. When $f$ is the indicator function of a non empty compact convex set, then an argument similar to the preceding shows that $\chi_n(f) = 1 $. Thus, we may define a measure $\mu_0(G) = \chi_n(f) $, where $G$ is any finite union of compact convex sets, and $f$ is the indicator function of the set $G$. We have thus proved the existence of a measure $\mu_0$ which is defined on all finite unions of compact convex sets, and which takes the value one on all non empty compact convex sets. This measure has a long history: it is the Euler characteristic. \bigskip Now you are thinking: if this is the Euler characteristic, then it is up to you to show that it coincides with what we ordinarily believe to be the Euler characteristic. Let us conclude this lecture by deriving the formula of Euler-Schl\"afli-Poincar\'e for polyhedra. As a matter of fact, this formula can be encapsulated into a simpler formula, one that is easy to remember. \bigskip Let $C$ be a non empty compact convex polytope of dimension $n$, and let $int(C)$ be the interior of $C$. Then we have the following fundamental formula for the Euler characteristic of $int(C)$: $$ \mu_0(int(C)) = (-1)^n .$$ Indeed, if $f$ is the indicator function of the set $int(C)$, we have: $$ \mu_0(int(C)) = \sum \chi_{n-1}(f_{\omega}) - \chi_{n-1}(f_{\omega + }) ,$$ where the sum ranges over all points $\omega$ on the line $L$ as above. But by induction, we see that every term on the right hand side equals zero, except when $\omega$ is the first point on the line $L$ for which the intersection $C \cap H_{\omega} $ is not empty. If $\omega_{\ell} $ is such a first point, then we have $$ \chi_{n-1}(f_{\omega_{\ell}}) = 0 $$ because the point $\omega_{\ell} $ is on the boundary of $C$, and $$ \chi_{n-1}(f_{\omega_{\ell} +}) = (-1)^{n-1} $$ by the induction hypothesis, because $ f_{\omega_{\ell} +}$ is the indicator function of the set $int(C) \cap H_{\omega_{\ell}+} $, which is the interior of a convex polyhedron one dimension lower. Putting all this together, we obtain $$ \mu_0(int(C)) = \sum \chi_{n-1}(f_{\omega_{\ell}}) - \chi_{n-1}(f_{\omega_{\ell}+}) = - (-1)^{n-1} = (-1)^n ,$$ as desired. We are now in a position to state the famous Euler formula for polyhedra. What is a polyhedron? A polyhedron is a finite union of convex polyhedra. Given a polyhedron, we must define a system of faces (of all dimensions, ranging from dimension $0$ (a point) to dimension $n$). We will say that a set $\bf{F}$ of convex polyhedra is a system of faces for an arbitrary polyhedron $K$ when the elements of $\bf{F}$, called faces, are non empty compact convex sets $F$ with disjoint interiors such that $$ K = \bigcup_{F \in \bf{F} }int(F). $$ Caution: the interior of a face of dimension $k$ is to be taken relative to the linear space of dimension $k$ that contains the face, and the interior of a point is a point. \bigskip Under these conditions we may take the Euler characteristic of both sides, and using the fact that any two interiors of faces are disjoint we obtain (using the fact that the measure of the disjoint union of a family of sets equals the sum of the measures of the individual sets): $$ \mu_0(K) = \sum_{F \in \bf{F} } \mu_0(int(F)) = f_0 - f_1 + f_2 - \dots + \dots , $$ where $f_i$ equals the number of faces of dimension $i$. This is Euler's formula. \bigskip We can now answer the second of the questions we had left open: how to extend the definition of the intrinsic volumes from compact convex sets to all finite unions of compact convex sets. If $G$ is such a finite union of compact convex sets, then we set $$ \mu_k (G) = \int \mu_0(G \cap \omega ) d\lambda_{n-k}^n(\omega), $$ where $\omega$ ranges over all linear varieties of dimension $n-k$ in $n$-space. The left hand side defines a measure, and when $G$ is a compact convex set it agrees with the definition we have already given. It is therefore the desired extension. The Euler characteristic does all the work for us. \bigskip We are now in a position to state the main theorem of geometric probability. We will say that an invariant measure $\mu$ on Euclidean $n$-space, defined on all finite unions of compact convex sets, is continuous, when $$ \lim_{C_n \rightarrow C} \mu(C_n) = \mu(C) $$ for all sequences $C_n$ of compact convex sets converging to the compact convex set $C$. We have the \bigskip {\bf Main Theorem of Geometric Probability} \bigskip The $n+1$ intrinsic volumes $\mu_0, \mu_1,\dots , \mu_n $ are a basis of the space of all continuous invariant measures defined on all finite unions of compact convex sets. \bigskip The first proof of this theorem is due to Hadwiger; the first elementary proof was published last year by Dan Klain of Georgia Tech. \bigskip In closing, let me try to answer the question you are about to ask: what has this got to do with geometric probability, anyway? \bigskip I will attempt a sketchy answer. Consider two compact convex sets $A$ and $B$. We imagine $B$ to be fixed in $n$-space, and that we "drop" the rigid set $A$ at random. What is the probability that $A$ meets $B$? We answer this question in three steps. First, we realize that by keeping $B$ fixed and varying $A$ by the group of Euclidean Motions, we define an invariant measure on convex sets $B$. Second, we apply Hadwiger's theorem, and infer that such an invariant measure equals a linear combination of the $n+1$ intrinsic volumes, with coefficients depending on $A$ and not on $B$. Third, we determine these coefficients by taking suitable $B$'s. The end result is an identity which is known as the kinematic formula, which has been the object of much research in this century, still going on today. \bigskip Thank you for your attention. \bigskip {\bf Bibliography} \bigskip Daniel A. Klain and Gian-Carlo Rota, Introduction to Geometric Probability (Lezioni Lincee), Cambridge University Press, 1997 \newpage \begin{center} {\bf INVARIANT THEORY, OLD AND NEW}\\ being\\ The second Colloquium Lecture\\ \bigskip delivered at the Annual Meeting of the American Mathematical Society\\ Baltimore, January 8, 1997.\\ \bigskip Gian-Carlo Rota\\ Department of Mathematics\\ MIT, room 2-351\\ 77 Massachusetts Avenue\\ Cambridge MA 02139-4307\\ \end{center} \bigskip \bigskip Invariant theory is the great Romantic story of mathematics. For one hundred and fifty years, from its beginnings with Boole to the time, around the middle of this century, when it branched off into several independent disciplines, mathematicians of all countries were brought together by their common faith in invariants: in England, Cayley, MacMahon, Sylvester and Salmon, and later, Alfred Young, Aitken, Littlewood and Turnbull. In Germany, Clebsch, Gordan, Grassmann, Sophus Lie, Study; in France, Hermite, Jordan and Laguerre; in Italy, Capelli, Brioschi, Trudi and Corrado Segre, in America, Glenn, Dickson, Carus (of the Carus Monographs), Eric Temple Bell and later Hermann Weyl. Seldom in history has an international community of scholars felt so united by a common scientific ideal for so long a stretch of time. In our century, Lie theory and algebraic geometry, differential algebra and algebraic combinatorics are offsprings of invariant theory. No other mathematical theory, with the exception of the theory of functions of a complex variable, has had as deep and lasting an influence on the development of mathematics. \bigskip Eventually, invariant theory was to become a victim of its own success: the very term "invariant theory" is nowadays understood in such a wide variety of senses that it has become all but meaningless. It is no wonder that you are baffled by the title of this lecture, and curious to hear what will be said about invariant theory in the next forty-eight minutes. \bigskip Like the Arabian phoenix arising from its ashes, classical invariant theory, once pronounced dead, is once again at the forefront of mathematics. The old treatises are being dusted off the shelves of library basements and reread, reinterpreted and presented in a language that meets the standard of rigor of our day. The program of classical invariant theory, that had for some time been given up as hopeless, is again being pursued, and success may at last be within reach. \bigskip We will review two turning points in the history of invariant theory. The first, the "new" one, happened around the turn of the century, and its effects are still being felt all over mathematics. The second, the "old" one, happened very early in the game, and led to a serious misunderstanding that lasts to this day. \bigskip A pedestrian definition of invariant theory might go as follows: invariant theory is the study of orbits of group actions. Such a definition is correct, but it must be supplemented by a programmatic statement. Hermann Weyl, in the introduction to his book "The Classical Groups", was the first in this century to give a sweeping overview of the program of invariant theory. He summarized this program in two basic assertions. The first states that "All geometric facts are expressed by the vanishing of invariants", and the second states that "all invariants are invariants of tensors". \bigskip Let us briefly comment on these lofty statements. What is a geometric fact? A geometric fact is a fact about space that is independent of the choice of a coordinate system. Geometric facts are described by means of equations which require a choice of coordinates. In a vector space $V$ of dimension $n$ one chooses a coordinate system $ x_1, x_2,\dots ,x_n $ . Since Descartes, we have learned to express geometric facts by equations in the coordinates $ x_1, x_2,\dots ,x_n $. However, about one hundred years ago, mathematicians and physicists made the shocking discovery that the usual type of equations, that is, equations in the commutative ring generated by the variables $ x_1, x_2,\dots ,x_n $, are inadequate for the description of a lot of geometric and physical facts. Motivated by this discovery, they introduced a more general ring. This is the ring of non commutative polynomials in the coordinates $ x_1, x_2,\dots ,x_n $. Homogeneous elements of this ring, that is, homogeneous non commutative polynomials in the variables $ x_1, x_2,\dots ,x_n $, are called tensors. If we believe Hermann Weyl's philosophy, then we will be satisfied that equations in the tensor algebra suffice for the description of any geometric fact we will ever meet. Furthermore, if these equations are to express geometric properties, then they must hold no matter what coordinate system is chosen; in other words, equations that describe geometric facts must be invariant under changes of coordinates. The program of invariant theory, from Boole to our day, is precisely the translation of geometric facts into invariant algebraic equations expressed in terms of tensors. \bigskip This program of translation of geometry into algebra was to be carried out in two steps. The first step consisted in decomposing tensor algebra into irreducible components under changes of coordinates. The second step consisted in devising an efficient notation for the expression of invariants for each irreducible component. The first step was successfully carried out in this century; the second was abandoned sometime in the twenties and only recently has it resurfaced. \bigskip The decomposition of tensor algebra into irreducible components was discovered around the turn of the century almost simultaneously by Issai Schur and Alfred Young. The gist of this decomposition is one of the great advances in mathematics of all times, and it may be worthwhile to present it in a form that can be made available to undergraduates. \bigskip Let us consider functions of three variables, such as $f(x_1, x_2, x_3)$. Two well known classes of functions of three variables are symmetric functions, defined to satisfy the equations $$f_s(x_1,x_2,x_3) = f_s(x_{i_1},x_{i_2},x_{i_3}) $$ for every permutation sending the indices $(1,2,3)$ to $(i_1,i_2,i_3)$, and skew symmetric functions, defined by the equations $$f_a(x_1,x_2,x_3) = \pm f_a(x_{i_1},x_{i_2},x_{i_3}), $$ where the sign is $+1$ or $-1$ according as the permutation sending the indices $(1,2,3)$ to $(i_1,i_2,i_3)$ is even or odd. \bigskip It is not true that a function of three variables is the sum of a symmetric function and a skew-symmetric function. A third type of function is required, which is called a cyclic function, which is defined by the equation $$f_c(x_1,x_2,x_3) + f_c(x_3,x_1,x_2) + f_c(x_2,x_3,x_1) = 0.$$ Every function of three variables can be uniquely written as the sum of a symmetric function, a skew symmetric function, and a cyclic function, in symbols: $$f(x_1,x_2,x_3) = f_s(x_1,x_2,x_3) +f_a(x_1,x_2,x_3) +f_c(x_1,x_2,x_3).$$ Each of the three symmetry classes is invariant under permutations; this fact is obvious for symmetric and skew symmetric functions but not quite so obvious for cyclic functions. These three invariant subspaces play for the group of permutations of a set of three elements a role analogous to the role of the eigenvectors of a symmetric matrix. \bigskip For functions $f(x_1,x_2,x_3,x_4) $ of four variables there are five symmetry classes, which are defined as follows: \bigskip 1. Symmetric functions. \bigskip 2. Skew symmetric functions. \bigskip 3. Cyclic symmetric functions, satisfying the four equations $$f(x_1,x_2,x_3,x_4)+ f(x_1,x_4,x_2,x_3)+f(x_1,x_3,x_4,x_2) = 0,$$ $$f(x_1,x_2,x_3,x_4)+f(x_4,x_2,x_1,x_3)+f(x_3,x_2,x_4,x_1) = 0,$$ $$f(x_1,x_2,x_3,x_4)+f(x_4,x_1,x_3,x_2)+f(x_2,x_4,x_3,x_1) = 0,$$ $$f(x_1,x_2,x_3,x_4)+f(x_3,x_1,x_2,x_4)+f(x_2,x_3,x_1,x_4)=0.$$ \bigskip 4. Functions satisfying the four equations $$f(x_1,x_2,x_3,x_4)+f(x_2,x_1,x_3,x_4)+f(x_1,x_2,x_4,x_3)+ f(x_2,x_1,x_4,x_3) = 0,$$ $$f(x_1,x_2,x_3,x_4)+f(x_3,x_2,x_1,x_4)+f(x_1,x_4,x_3,x_2)+ f(x_3,x_4,x_1,x_2)= 0,$$ $$f(x_1,x_2,x_3,x_4)+f(x_1,x_3,x_2,x_4)+f(x_4,x_2,x_3,x_1)+ f(x_4,x_3,x_2,x_1)= 0,$$ $$\sum sign(\sigma ) f(x_{\sigma 1},x_{\sigma 2},x_{\sigma 3}, x_{\sigma 4})= 0.$$ \bigskip 5. Functions satisfying the equations $$f(x_1,x_2,x_3,x_4)-f(x_2,x_1,x_3,x_4)-f(x_1,x_2,x_4,x_3)+ f(x_2,x_1,x_4,x_3) = 0,$$ $$f(x_1,x_2,x_3,x_4)-f(x_3,x_2,x_1,x_4)-f(x_1,x_4,x_3,x_2)+ f(x_3,x_4,x_1,x_2)= 0,$$ $$f(x_1,x_2,x_3,x_4)-f(x_1,x_3,x_2,x_4)-f(x_4,x_2,x_3,x_1)+ f(x_4,x_3,x_2,x_1)= 0,$$ $$\sum f(x_{\sigma 1},x_{\sigma 2},x_{\sigma 3}, x_{\sigma 4})= 0.$$ \bigskip Every function of four variables is uniquely expressible as the sum of five functions, each one belonging to one of these symmetry classes. Each symmetry class is invariant under permutations. \bigskip More generally, every function of $n$ variables $f(x_1,x_2,\dots ,x_n)$ can be uniquely written as the sum of $p_n$ functions, each one belonging to a different symmetry class. Here, $p_n$ equals the number of partitions of the integer $n$. Each symmetry class is defined by equations which are not difficult to find. \bigskip This decomposition holds for tensors as well, after some cosmetic changes of notation. To this day, only two symmetry classes of tensors have been studied in any detail. Symmetric tensors are ordinary commutative polynomials such as we learned to use in analytic geometry. Skew symmetric tensors are polynomials in the coordinates $ x_1, x_2,\dots ,x_n $ provided that the variables are assumed to satisfy the equations $x_ix_j = - x_jx_i $. Tensors belonging to symmetry classes other than the classes of symmetric and skew symmetric tensors also occur in geometry and physics. However, these symmetry classes have been studied very little, and they are a long way from being understood. \bigskip So much for the word "new" in the title of this lecture; let us next do some justice to the word "old". We will describe the most peculiar feature of classical invariant theory, namely, the symbolic or umbral notation, to which Eric Temple Bell dedicated his Colloquium Lectures in 1927. We will consider the simplest group, namely, the group of translations of the line. The unusual features of the symbolic method will already be apparent in this special case. Let $p(x)$ and $ q(x) $ be monic polynomials in the variable $x$. We write them in the following quaint notation: $$ p(x) = x^n + {n \choose 1}a_1x^{n-1} + {n \choose 2}a_2 x^{n-2} + \dots + {n \choose n-1}a_{n-1}x + a_n , $$ and $$ q(x) = x^k + {k \choose 1}b_1x^{k-1} + {k \choose 2}b_2 x^{k-2} + \dots + {k \choose k-1}b_{k-1}x + b_k . $$ We assume that the polynomial $q(x)$ is of lower degree than the polynomial $p(x)$, that is, that $k \leq n $. \bigskip Define the translation operator $T^c$ on a polynomial $p(x)$ as follows: $$ T^c p(x) = p(x + c) .$$ Let us write $$ p(x+c) = x^n +{n \choose 1}p_1(c) x^{n-1} + {n \choose 2}p_2(c) x^{n-2} + \dots + {n \choose n-1}p_{n-1}(c) x + p_n(c) , $$ The $j$-th coefficient $p_j(c)$ of the polynomial $p(x+c) $ is computed to be $$ p_j(c) = a_j + {j \choose 1} a_{j-1} c + {j \choose 2 }a_{j-2}c^2 + \dots + c^j . $$ A polynomial I($a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k) $ in the variables $a_1$, $a_2$, \dots ,$ a_n$, $b_1$,$ b_2$, \dots ,$ b_k $ is said to be an invariant of the two polynomials $p(x)$, $q(x)$ when $$ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k) = I(p_1(c),p_2(c), \dots , p_n(c), q_1(c), q_2(c), \dots , q_k(c) ) $$ for all complex numbers $c$. By abuse of notation, we write $I(p(x), q(x)) $ and we speak of $I$ as being an invariant of the polynomials $p(x)$ and $q(x)$. In this abusive notation, a polynomial $I$ is said to be an invariant of the polynomials $p(x)$ and $q(x)$ whenever $$ E(T^cp(x), T^cq(x) ) = E(p(x), q(x)) $$ for all constants $c$. \bigskip Invariant theory is concerned with the problem of finding all invariants of a given set of polynomials, as well as their significance. \bigskip What is meant by the "significance" of an invariant? We will appeal to Hermann Weyl. "Every" property of polynomials which is invariant under the group of translations is expressed by the vanishing of a set of invariants. In other words, "any" set of polynomials which is invariant under translations is the same set as a set of polynomials obtained by setting to zero a set of invariants of such polynomials. \bigskip It is impossible to understand the above statement without examples. Let us consider the simplest and oldest example. The property of a quadratic polynomial $$ q(x) = x^2 + 2b_1 x + b_2 .$$ of having a double root is invariant under translations; in other words, if the polynomial $q(x)$ has a double root, so does the polynomial $q(x+c)$ for any constant $c$ . Following Hermann Weyl, we look for an invariant whose vanishing expresses this property. Sure enough, it is easy to check that the discriminant $$ D( b_1, b_2) = b_1^2 - b_2 $$ is the desired invariant. This example, due to Boole, was the spark that led to the birth of invariant theory. \bigskip One often hears the sentence "Hilbert killed invariant theory", repeated as an excuse to ignore all that went on in invariant theory after Hilbert. I don't know who made up this infamous sentence. It is not true. Hilbert loved invariant theory, and he went on publishing striking papers in invariant theory well after he proved the theorem that is nowadays called the Hilbert basis theorem, the theorem that is supposed to have killed invariant theory. Some of the most fascinating results in invariant theory were discovered in the first twenty years of this century, a long time after Hilbert proved his basis theorem. \bigskip What then is the reason for the temporary demise of invariant theory in this century? One reason is the endemic use of a notation that lacked rigor and that amounted to little more than handwaving in print. This is the symbolic or umbral notation. \bigskip Dieudonn\'e wrote that half the success of a piece of mathematics depends on a proper choice of notation. It would be interesting to make a list of unfortunate notations that killed various chapters of mathematics, as well as a list of felicitous notations that promoted the development of other branches of mathematics. The symbolic or umbral notation that was used by invariant theorists through the nineteen twenties was a catastrophe. A number of mathematicians tried to make sense of the symbolic method without success, the three most notable ones being Hermann Weyl, Eric Temple Bell, and Edward Hegeler Carus. Eric Temple Bell failed to properly define umbral notation, and his book "Algebraic Arithmetic" remains to this day the book of seven seals. If Hermann Weyl and Eric Temple Bell had lived fifty years longer, so as to benefit of the development of what was in their time called "modern" algebra, they would undoubtedly have succeeded in properly defining umbral notation. \bigskip In our day, it does not take much work to accomplish this task. Do not be alarmed: it will only take a few minutes. Before I start spouting out definitions, let me say what I am not going to say. Umbral notation can be shown to be equivalent, or "cryptomorphic", to use a term invented by my late friend Garrett Birkhoff, to another notation that has gained great notoriety in our day: I mean the notation of Hopf algebras. I will not justify this Sybilline pronouncement, not because it is difficult to do so, but because it would be too boring to do so. \bigskip Let us go on to the definition of umbral notation. \bigskip Side by side with the polynomials $p(x)$ and $q(x)$, we consider another polynomial algebra ${\bf C}[ x, \alpha , \beta ] $ in three variables $x$ , $ \alpha $ and $\beta $, together with a linear functional $E$ defined on the underlying vector space ${\bf C}[x, \alpha, \beta ]$. The definition of the linear functional $E$ is the key point. It is carried out in the following steps: \bigskip Step 1. Set $$ E(x^j ) = x^j $$ for all non negative integers $j$ , in particular $E(1) = 1$. Thus, the range of the linear functional $E$ is $C[x]$. \bigskip Step 2. Set $$ E(\alpha^j) = a_j ,$$ in particular, we have $ E(\alpha^j) = 0 $ if $j > n $. \bigskip Step 3. Set $$ E(\beta^j) = b_j ,$$ in particular, we have $ E(\beta^j) = 0 $ if $j > k $. \bigskip Step 4. This is the main step. Set $$ E(\alpha^i \beta^j x^{\ell}) = E(\alpha^i ) E( \beta^j )x^{\ell}. $$ Following Sylvester, the variables $\alpha $ and $\beta $ are called umbrae. In other words, the linear functional $E$ is multiplicative on distinct umbrae. \bigskip Step 5. Extend by linearity. \bigskip This completes the definition of the linear functional $E$. We next come to the most disquieting feature of umbral notation. Let $f(\alpha , \beta , x) $ and $g(\alpha , \beta , x) $ be two polynomials in the variables $ \alpha , \beta , x $. We write $$ f(\alpha , \beta , x) \cong g(\alpha , \beta , x) $$ to mean $$ E(f(\alpha , \beta , x)) = E( g(\alpha , \beta , x) ) . $$ Read $\cong $ as "equivalent to". The "classics" went a bit too far, they wrote $$ f(\alpha , \beta , x) = g(\alpha , \beta , x) $$ that is, they replaced the symbol $\cong$ by ordinary equality. This was an excessive abuse of notation. The "classics" were aware of the error, and while they avoided computational errors by clever artistry, they were unable to settle on a correct notation. \bigskip The umbral or symbolic method consists of replacing all occurrences of the coefficients of the polynomials $p(x)$ and $q(x)$ by umbrae and equivalences. For example, $$ p(x) \cong (x + \alpha)^n $$ and $$ q(x) \cong (x+ \beta)^k .$$ Let us carefully check the first equivalence. By definition, the equivalence means the same as $$ E(p(x)) = E( (x + \alpha)^n).$$ Since $E(x^j) = x^j $ for all non negative integers $j$, this identity can be rewritten as $$ p(x) = E( (x + \alpha)^n).$$ Expanding the right hand side by the binomial theorem, we obtain $$ E( (x + \alpha)^n) = $$ $$ E( x^n + {n \choose 1}\alpha x^{n-1} + {n \choose 2}\alpha^2 x^{n-2} + \dots + {n \choose n-1}\alpha^{n-1}x +\alpha^n) $$ By linearity this equals $$ x^n + {n \choose 1}E(\alpha) x^{n-1} + {n \choose 2}E(\alpha^2) x^{n-2} + \dots + {n \choose n-1}E(\alpha^{n-1})x + E(\alpha^n) . $$ Evaluating the linear functional $E$, we see that this in turn equals $$ x^n + {n \choose 1}a_1x^{n-1} + {n \choose 2}a_2 x^{n-2} + \dots + {n \choose n-1}a_{n-1}x + a_n , $$ as desired. \bigskip The expression $$ (x + \alpha)^n $$ is called an umbral representation of the polynomial $p(x)$. In umbral notation, a complex number $r$ is a root of the polynomial equation $p(x) = 0$ if and only if $$ (r + \alpha)^n \cong 0. $$ Similarly, in umbral notation the polynomial $T^cp(x)= p(x+c)$ may be represented as follows: $$ p(x+c ) \cong (x+\alpha + c)^n, $$ and this yields the umbral expression for the coefficients $p_j(c)$ of the polynomial $p(x+c)$, namely $$ p_j(c) \cong (\alpha + c)^j . $$ Let us next see how umbral notation is related to invariants. Let us assume that the two polynomials $p(x)$ and $q(x)$ have the same degree $n$. Then an invariant $A$ of the polynomials $p(x), q(x)$ may be defined as follows: $$ A(q(x), p(x)) \cong (\beta - \alpha)^n .$$ The evaluation of the invariant $A$ in terms of the coefficients of $p(x)$ and $q(x)$ proceeds as follows: $$ A(q(x), p(x)) = E((\beta - \alpha)^n) = $$ $$E(\beta^n -{n \choose 1} \beta^{n-1}\alpha + \dots + (-1)^{n-1}{n \choose n-1}\beta \alpha^{n-1} + (-1)^n \alpha^n) = $$ $$ E(\beta^n) - E({n \choose 1} \beta^{n-1}\alpha) + \dots + (-1)^{n-1}E({n \choose n-1}\beta \alpha^{n-1}) + (-1)^n E(\alpha^n)= $$ $$ E(\beta^n) - {n \choose 1} E(\beta^{n-1})E(\alpha) + \dots + (-1)^{n-1}{n \choose n-1}E(\beta) E(\alpha^{n-1}) + (-1)^n E(\alpha^n) = $$ $$ b_n - {n \choose 1} b_{n-1}a_1 + {n \choose 2}b_{n-2}a_2 - \dots + \dots + (-1)^{n-1}{n \choose n-1}b_1a_{n-1} + (-1)^n a_n. $$ Why is $A$ an invariant? This is best seen in umbral notation: $$ A(T^cq(x), T^cp(x)) \cong (\beta + c - \alpha - c )^n = (\beta - \alpha)^n. $$ The invariant $A$ is called the apolar invariant; two polynomials $p(x)$ and $q(x)$ having the property that $A(q(x), p(x)) = 0 $ are said to be apolar. In umbral notation, two polynomials are apolar whenever $$ (\beta - \alpha)^n \cong 0. $$ The concept of apolarity has a distinguished pedigree going all the way back to Apollonius. \bigskip What is the "significance" of the apolar invariant? What does it mean for two polynomials to be apolar? This question is answered by \bigskip {\bf Theorem 1}. Suppose that $r$ is a root of the polynomial $q(x)$, that is, that $q(r)= 0$. Then the polynomials $q(x)$ and $p(x) = (x - r)^n $ are apolar. \bigskip Proof. For $p(x) = (x-r)^n $ we have $\alpha^j \cong (-r)^j $, and hence $$ A(q(x),p(x)) \cong (\beta - (-r))^n = (\beta + r)^n \cong 0, $$ as desired. \bigskip {\bf Corollary}. If the polynomial $q(x)$ has $n$ distinct roots $r_1,r_2,\dots , r_n $, and if the polynomial $p(x)$ is apolar to $q(x)$, then there exist constants $c_1,c_2,\dots ,c_n $ for which $$ p(x) = c_1 (x-r_1)^n + c_2 (x-r_2)^n + \dots + c_n (x-r_n)^n . $$ \bigskip Proof. The dimension of the affine subspace of all monic polynomials $p(x)$ which are apolar to $q(x)$ equals $n$. But if the polynomial $q(x)$ has simple roots, then by the above theorem the polynomials $(x-r_1)^n, (x-r_2)^n,\dots , (x- r_n)^n $ are linearly independent and apolar to $q(x)$. Hence the polynomial $p(x)$ is a linear combination of these polynomials. This completes the proof. \bigskip Thus, we see that apolarity gives a trivial answer to the following question: when can a polynomial $p(x)$ be written as a linear combination of polynomials of the form $(x- r_1)^n, (x-r_2)^n,\dots , (x-r_n)^n $? \bigskip A beautiful theorem on apolarity was proved by the British mathematician John Hilton Grace. We state it without proof: \bigskip {\bf Grace's Theorem}. If two polynomials $p(x)$ and $q(x)$ of degree $n$ are apolar, then every disk in the complex plane containing every zero of $p(x)$ also contains at least one zero of $q(x)$. \bigskip Grace's Theorem is an instance of what might be called a sturdy theorem. For almost one hundred years it has resisted all attempts at generalization. Almost all known results about the distribution of zeros of polynomials in the complex plane are corollaries of Grace's theorem. \bigskip We will next generalize the apolar invariant to the case of two polynomials $p(x)$ and $q(x)$ of different degrees $n$ and $k$, with $ k \leq n$. To this end, we slightly generalize the definition of invariant, as follows. \bigskip A polynomial $ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k, x ) $ in the variables $a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k , x $ is said to be an invariant of the polynomials $p(x)$, $q(x)$ when $$ I(a_1,a_2, \dots , a_n , b_1, b_2, \dots , b_k, x) = $$ $$I(p_1(c),p_2(c), \dots , p_n(c), q_1(c), q_2(c), \dots , q_k(c), x+ c ) $$ for all complex numbers $c$. \bigskip Sometimes these more general invariants are called covariants. \bigskip We define a more general apolar invariant as follows: $$ A(q(x), p(x)) \cong (\beta - \alpha)^k (x- \alpha )^{n-k}. $$ Again, we say that two polynomials $p(x)$ and $q(x)$ are apolar when $A(q(x), p(x))$ is identically zero, that is, zero for all $x$. Theorem 1 remains valid as stated. That is, if $q(r) = 0$ then the polynomial $p(x) = (x-r)^n $ is apolar to $q(x)$. \bigskip Let us consider a special case. Suppose that $q(x)$ is a quadratic polynomial and $p(x)$ is a cubic polynomial: $$ q(x) = x^2 + 2b_1 x + b_2 $$ and $$ p(x) = x^3 + 3 a_1 x^2 + 3a_2 x + a_3 .$$ Then we have, in umbral notation $$ A(q(x), p(x)) \cong (\beta - \alpha)^2 (x - \alpha) = \\ (\beta^2 - 2 \alpha \beta + \alpha^2)x - \alpha \beta^2 + 2 \alpha^2 \beta - \alpha^2. $$ Evaluating the linear functional $E$, we obtain the following explicit expression for the apolar invariant: $$ A(q(x), p(x)) = E((\beta^2 - 2 \alpha \beta + \alpha^2)x - \alpha \beta^2 + 2 \alpha^2 \beta - \alpha^2) =$$ $$ E (\beta^2 ) - 2E( \alpha \beta) +E( \alpha^2)x +E ( - \alpha \beta^2) + 2 E(\alpha^2 \beta) -E( \alpha^2) =$$ $$E (\beta^2 ) - 2E( \alpha)E( \beta) +E( \alpha^2)x +E ( - \alpha)E( \beta^2) + 2 E(\alpha^2)E( \beta) -E( \alpha^2) = $$ $$ (b_2 - 2 a_1b_1 + a_2)x - a_1 b_2 + 2 a_2 b_1 - a_3. $$ Thus, a quadratic polynomial $q(x)$ and a cubic polynomial $p(x)$ are apolar if and only if their coefficients satisfy the two equations $$ b_2 - 2 a_1b_1 + a_2 = 0 $$ $$ - a_1 b_2 + 2 a_2 b_1 - a_3 = 0 . $$ Using these equations, we can prove two important theorems: \bigskip {\bf Theorem 2}. There is in general one quadratic polynomial which is apolar to a given cubic polynomial. \bigskip Proof. Indeed, the above equations may be rewritten as $$ b_2 - 2 a_1b_1 = - a_2 $$ $$ - a_1 b_2 + 2 a_2 b_1 = a_3. $$ The solutions $b_1, b_2$ for given $a_1, a_2, a_3 $ are in general unique. \bigskip {\bf Theorem 3}. There is always a two-dimensional space of cubic polynomials which are apolar to a given quadratic polynomial. \bigskip Proof. Indeed, given $b_1, b_2$ we may solve for $a_1, a_2, a_3 $ from the equations $$- 2 a_1b_1 + a_2 = - b_2 $$ $$ - a_1 b_2 + 2 a_2 b_1 = a_3 .$$ These equations always have a double infinity of solutions, as they used to say in the old days. \bigskip Theorems 2 and 3 provide a simple and explicit method for solving a cubic equation. It goes as follows. \bigskip Given the cubic polynomial $$ p(x) = x^3 + 3 a_1 x^2 + 3a_2 x + a_3 ,$$ first, by Theorem 2 we find a unique quadratic polynomial $q(x)$ which is apolar to $p(x)$. In general, such a quadratic polynomial $q(x)$ has two distinct roots $r_1$ and $r_2$. By Theorem 1, the cubic polynomials $ (x-r_1)^3$ and $(x-r_2)^3$ are apolar to $q(x)$. Second, by Theorem 3, the affine linear space of cubic polynomials apolar to $q(x)$ has dimension two. Since $p(x)$ is apolar to $q(x)$, we conclude that $p(x)$ is a linear combination of $ (x- r_1)^3$ and $(x-r_2)^3$. In symbols: $$ p(x) = c (x-r_1)^3 + (1-c) (x-r_2)^3 $$ for some constant $c$. Observe that $c$, $r_1$ and $r_2$ are computed by solving linear and quadratic equations. \bigskip In this way, the solution of the cubic equation $p(x) = 0$ is reduced to the solution of the equation $$ c (x-r_1)^3 = - (1-c) (x-r_2)^3 ,$$ and this equation is easily solved by taking a cube root. \bigskip This method of solving a cubic equation is the only one I can remember. \bigskip Let me digress with a personal anecdote. A few years ago, I was lecturing on this material at a symposium in combinatorics that took place at the University at Minnesota. Persi Diaconis was sitting in the front row, and I could tell as I started to lecture that he was falling asleep; he eventually began to doze off. But the moment I mentioned the magic words "solving a cubic equation" he woke up with a start and said: "Really! How?" \bigskip The preceding two theorems are easily generalized. \bigskip {\bf Theorem 4}. The dimension of the space of all (monic) polynomials of degree $k$ that are apolar to a polynomial of degree $n$ equals $2k-n$, in general, when $ k \leq n $. \bigskip {\bf Theorem 5}. The dimension of the space of all (monic) polynomials of degree $n$ that are apolar to a polynomial of degree $k$ equals $k$, if $k \leq n$. \bigskip Let us try to solve an equation of degree 5 in much the same way as we solved a cubic equation. Given the quintic polynomial $$ p(x) = x^5 + 5a_1x^4 + 10 a_2 x^3 + 10 a_3 x^2 + 5 a_4 x + a_5 = 0 ,$$ Theorem 4 assures us that there is in general a unique cubic polynomial $q(x)$ which is apolar to $p(x)$. In general, this cubic polynomial has three distinct roots $r_1, r_2, r_3$. By Theorem 1, the polynomials $(x-r_1)^5, (x-r_2)^5, (x-r_3)^5 $ are linearly independent and apolar to $q(x)$ . By Theorem 5, the dimension of the space of all polynomials apolar to $q(x)$ equals $3$. But the polynomial $p(x)$ is apolar to $q(x)$. Hence, $p(x)$ can be written in the form $$ p(x) = c_1(x-r_1)^5 + c_2(x-r_2)^5 + c_3(x - r_3)^5 $$ for suitable constants $c_i$. Thus, we see that a generic polynomial of degree 5 can be written as a linear combination of three fifth powers of linear polynomials. These are computed by solving linear, quadratic and cubic equations. This reduction to canonical form of the quintic is as close as we can come to solving a quintic equation by radicals. \bigskip At this point, someone in the audience will raise his or her hand and say: "Excuse me, but the umbral method you have introduced is not even good enough to express the discriminant of a quadratic equation! " \bigskip Quite right. \bigskip The definitions of umbrae and of the linear functional $E$ have an obvious generalization to any array of polynomials, say $p_1(x), p_2(x),\dots , p_{\ell}(x) $. One simply considers the space of polynomials $$ {\bf C}[x, \alpha_1 ,\alpha_2, \dots , \alpha_{\ell} ] $$ and one sets $$E(\alpha_{t}^j )$$ to equal the $j$-th coefficient of the polynomial $p_{t}(x)$. What is crucial, the linear functional $E$ is again multiplicative on distinct umbrae: $$ E(\alpha_1^i \alpha_2^j \alpha_3^k \dots x^{\ell} ) = E(\alpha_1^i )E(\alpha_2^j) E( \alpha_3^k )\dots x^{\ell}.$$ Now comes the catch that in the old days was to turn into a notational nightmare: the polynomials say $p_1(x), p_2(x),\dots , p_{\ell}(x) $ need not be distinct. In fact, the most important case occurs when each of the polynomials $p_1(x), p_2(x),\dots , p_{\ell}(x) $ is equal to one and the same polynomial $p(x)$. In this case, the definition of the linear functional $E$ may be simplified as follows: 1. $$E(\alpha_i^j) = a_j $$ for every $i$ , and 2. $$ E(\alpha_1^i \alpha_2^j \alpha_3^k \dots x^{\ell}) = a_i a_j a_k \dots x^{\ell} $$ for all non negative integers $i,j, k, \dots \ell $. \bigskip Umbrae $ \alpha_1 ,\alpha_2, \dots , \alpha_{\ell} $ satisfying 1 and 2 are said to be exchangeable. Thus, for exchangeable umbrae we have $$ (x + \alpha_1)^n \cong (x+\alpha_2)^n. $$ Eric Temple Bell, who wrote $=$ in place of $\cong$, was baffled by the fact that two umbrae could be exchangeable without being equal. \bigskip We can now state the main theorem of invariant theory. We will consider a single polynomial. \bigskip {\bf Theorem 6}. Every invariant of a polynomial $p(x)$ is obtained by evaluating some polynomial in the differences $\alpha_i - \alpha_j $ and $\alpha_i - x,$ where $\alpha_i $ and $\alpha_j $ are exchangeable umbrae. Conversely, every polynomial in such differences is equivalent to an invariant of the polynomial $p(x)$. \bigskip The proof is extremely simple, but will be omitted. \bigskip Let us review some classical examples. \bigskip The discriminant of a quadratic polynomial $p(x) = x^2 +2a_1x+a_2 $ may be umbrally represented as follows: $$ D(p(x)) \cong (\alpha_1 - \alpha_2)^2 / 2, $$ where $\alpha_1 $ and $\alpha_2 $ are exchangeable umbrae. Indeed: $$ E( (\alpha_1 - \alpha_2)^2 ) =E( \alpha_1^2) - 2 E(\alpha_1\alpha_2) + E(\alpha_2 ^2) =a_2 - 2 a_1^2 + a_2 = 2 (a_2 - a_1^2), $$ as desired. \bigskip Let us next consider a cubic polynomial $p(x) = x^3 + 3a_1 x^2 + 3a_2 x + a_3 $. The discriminant of this polynomial, let us call it $D(p(x)) $, equals, as you know, the expression $$ D(p(x)) = \frac{4(a_2 - a_1)^2(a_1a_3 - a_2^2) - (a_3 - a_1a_2)^2}{2} .$$ The umbral expression of the discriminant is easier to remember. $$ D(p(x)) \cong (\alpha_1 - \alpha_2)^2(\alpha_3 - \alpha_4)^2(\alpha_1 - \alpha_4)(\alpha_2 - \alpha_3). $$ As you know, the discriminant vanishes if and only if the cubic equation $p(x) = 0 $ has a double root. \bigskip The Hessian of a cubic polynomial can be elegantly written in umbral notation as follows: $$ H(p(x)) \cong (\alpha_1 - \alpha_2)^2(\alpha_1 - x)(\alpha_2 - x). $$ The Hessian vanishes if and only if the cubic polynomial is the third power of a polynomial of degree one. \bigskip Allow me another digression. On hearing about the vanishing of the Hessian as the condition that a cubic polynomial be a perfect cube, it comes naturally to ask the general question: which invariant of a polynomial of degree $n$ vanishes if and only if the polynomial is the $k$- th power of some polynomial of degree $n/k$? Here $k$ is a divisor of $n$. For a long time I thought the answer to this question to be beyond reach, until one day, while leafing despondently through the second volume of Hilbert's collected papers, I accidentally discovered that Hilbert had completely solved it. The solution can be elegantly expressed in umbral notation. This is only one of several striking results of Hilbert's on invariant theory that have been forgotten. \bigskip Let us consider next an invariant of the quintic. Theorem 3 tells us that a quintic $p(x) = x^5 + 5a_1x^4 + 10 a_2x^3 + 5 a_3x^4 + a_5 $ has a unique apolar cubic polynomial $q(x)$. The polynomial $q(x)$ is an invariant of $p(x)$. Does it have a simple expression in umbral notation? Indeed it does. The expression is the following: $$ q(x) \cong (\alpha_2 - \alpha_3)^2(\alpha_3 - \alpha_1)^2(\alpha_1 - \alpha_2)^2(\alpha_1 - x)(\alpha_2 - x)(\alpha_3 - x). $$ In the classical literature this invariant is denoted by the letter $j$. What property will the quintic polynomial $p(x)$ have when the invariant $j$ vanishes? The answer to this question is pleasing. The invariant $j$ of a quintic polynomial is identically equal to zero if and only if the quintic is apolar to some non trivial polynomial of degree two. But then Theorem 5 tells us that the quintic may be written in the form $$ p(x) = c(x-r_1)^5 + (1-c)(x-r_2)^5, $$ where $r_1, r_2 $ are the roots of a quadratic equation. Thus, the vanishing of the invariant $j$ is a necessary and sufficient condition that the quintic polynomial $p(x)$ may be written as the sum of two rather than three fifth powers of linear polynomials. When this is the case, the fifth degree equation $p(x) = 0$ can be solved by radicals. By similar arguments, one can compute all invariants whose vanishing implies that the equation of degree five is algorithmically solvable by radicals. Twenty-three invariants play a relevant role, as Cayley was first to show. \bigskip Hilbert's theorem on finite generation of the ring of invariants can be recast in the language of umbrae, and can be given a simple combinatorial proof that dispenses with the Hilbert basis theorem. \bigskip In closing, let us touch upon another reason for the demise of the symbolic method in invariant theory. \bigskip In mathematics, it is extremely difficult to tell the truth. The formal exposition of a mathematical theory does not tell the whole truth. The truth of a mathematical theory is more likely to be grasped while we listen to a casual remark made by some expert that gives away some hidden motivation, when we finally pin down the typical examples, or when we discover what the real problems are that were stored behind the showcase problems. \bigskip Philosophers and psychiatrists should explain why it is that we mathematicians are in the habit of systematically erasing our footsteps. Scientists have always looked askance at this strange habit of mathematicians, which has changed little from Pythagoras to our day. \bigskip The hidden purpose of the symbolic method in invariant theory was not simply that of finding easy expressions for invariants. A deeper faith was guiding this method. It was the expectation that the expression of invariants by the symbolic method would eventually guide us to single out the "relevant" or "important" invariants among an infinite variety. It was the hope that the significance of the vanishing of an invariant could be gleaned from its umbral expression. The vanishing of this faith was the real reason for the demise of classical invariant theory, and the revival of this faith is the reason for its present rebirth. \bigskip Whether or not we will succeed this second time where the classics failed is a cliffhanger that will probably be resolved in the next few years. I would not be speaking to you now if I did not believe in success. \bigskip Thank you for your attention. \bigskip {\bf Bibliography} \bigskip J.P.S. Kung and Gian-Carlo Rota, The invariant theory of binary forms, Bulletin of the American Mathematical Society, vol.10 (1984), pp. 27-85. \bigskip Frank D. Grosshans, Gian-Carlo Rota and Joel A. Stein, Invariant Theory and Superalgebras, CBMS Regional Conferences in Mathematics, vol. 69, American Mathematical Society,1987. \bigskip N. Metropolis and Gian-Carlo Rota, Symmetry classes: functions of three variables, American Mathematical Monthly, vol. 98 (1991), pp. 328-332. \bigskip N. Metropolis, G.-C. Rota and Joel A. Stein, Theory of symmetry classes, Proceedings of the National Academy of Sciences, vol. 88 (1991), pp. 8415-8419. \bigskip Richard Ehrenborg and Gian-Carlo Rota, Apolarity and canonical forms for homogeneous polynomials, European Journal of Combinatorics, vo. 14 (1993), pp. 157-181. \bigskip Gian-Carlo Rota and B. D. Taylor, The classical umbral calculus, SIAM Journal of Mathematical Analysis, vol. 25 (1994), pp.694-711. \bigskip Antonio Di Crescenzo and Gian-Carlo Rota, Sul calcolo umbrale, Ricerche di Matematica, vol XLIII (1994), pp. 129-162. \bigskip N. Metropolis, Gian-Carlo Rota and Joe. A. Stein, Symmetry classes of functions, Journal of Algebra, vo. 171 (1995), pp. 845-866. \newpage \begin{center} {\bf COMBINATORIAL SNAPSHOTS\\ being\\ The third of three Colloquium Lectures\\ delivered at the Annual Meeting of the American Mathematical Society\\ Baltimore, January 9, 1998\\ Gian-Carlo Rota\\ Department of Mathematics\\ MIT\\ Cambridge MA 02139-4307\\} \end{center} \bigskip \bigskip When I was in high school, my English teacher gave me to read an essay by James Thurber, called "The secret life of Walter Mitty". After rereading this essay every few years, I decided that everyone has a Walter Mitty complex. One way to understand a person might be to discover that person's Walter Mitty fantasies. Most of our mathematical thoughts in high school or in college were Walter Mitty fantasies. When we learned a new piece of math, we would find ourselves fantasizing on its possible generalizations. As soon as we understood binomial coefficients, we fantasized about their generalization to the case when the denominator is negative; the moment we learned about derivatives, we launched into derivatives of fractional order. If we were ever exposed to the Riemann zeta function, we would romanticize some new interpretation of this function that would give away its secret. \bigskip This lecture should have been given another title. It should be called "The later life of Walter Mitty". It will consist of a sequence of displays of "chutzpah" by a Walter Mitty who has lost his shyness. Each snapshot will deal with some youthful fantasy that has partially worked out. \newpage FIRST SNAPSHOT: AN EXAMPLE OF PROFINITE COMBINATORICS. \bigskip Let us begin with a piece of history-fiction, and fantasize how Riemann might have discovered the Riemann zeta function. \bigskip Professor Riemann was aware that arithmetic density is of fundamental importance in number theory. If $A$ is a subset of the set of positive integers $N$, then the arithmetic density of the set $A$ is defined to be $$ dens(A)=\lim_{n\rightarrow \infty} {1\over n} \|{A\cap \{1,2, \dots , n\}}\|.$$ \bigskip whenever the limit exists. For example $dens(N) = 1$. If $A_p$ is the set of multiples of the prime $p$, then $dens(A_p) = \frac{1}{p}$; what is more appealing, one easily computes that $dens (A_p \cap A_q) = \frac{1}{pq} $ for any two primes $p$ and $q$. If density were a (countably additive) probability measure, we would infer that the events that a randomly chosen number is divisible by either of two primes are independent. Unfortunately, arithmetic density shares some but not all properties of a probability measure. It is most emphatically not countably additive. \bigskip After a period of soul searching, Professor Riemann was able to find a remedy to some deficiencies of arithmetic density by a brilliant leap of imagination. He chose a real number $s > 1$, and defined the measure of a positive integer $n$ to equal $\frac{1}{n^s}$; in this way, the measure of the set $N$ turned out to equal $$ \zeta (s) = \sum_{n=1}^{\infty}\frac{1}{n^s}. $$ \bigskip Therefore, he could define a (countably additive) probability measure $P_s$ on the set $N$ of positive integers by setting $$ P_s(A) = \frac{1}{\zeta (s)} \sum_{n \in A }\frac{1}{n^s}. $$ Riemann then proceeded to verify what he had sensed all along, namely, the fundamental property $$P_s(A_p \cap A_q) = P_s(A_p)P_s(A_q) = \frac{1}{pq}. $$ In other words, the events $A_p$ and $A_q$ that a randomly chosen integer $n$ be divisible by one of the two primes $p$ or $q$ are independent relative to the probability $P_s $. The Riemann zeta function was good for something, after all. \bigskip I will now use a rhetorical device that was effectively employed by one of my undergraduate teachers, Professor Bochner. In the classroom, Professor Bochner would prefix the statement of a theorem by the words: "Subject to technical assumptions, the following is true"; without, of course, ever disclosing what his technical assumptions were. \bigskip Professor Riemann then proceeded to show that, subject to technical assumptions on the set $A$, $$ \lim_{s\rightarrow 1}P_s(A) = dens (A). $$ \bigskip Thus, even though arithmetic density is not a probability, it is under suitable conditions the limit of probabilities. \bigskip Long after Riemann was gone, it was shown, again subject to technical assumptions, that the probabilities $P_s$ are the only probabilities defined on the set $N$ of natural integers for which the events of divisibility by different primes are independent. This fact seems to lend support to the program of proving results of number theory by probabilistic methods based upon the Riemann zeta function. \bigskip Why didn't Professor Riemann ever publish this wonderful idea of his? The answer is not hard to find. True, some theorems of number theory can be proved probabilistically by this limiting process, for example Dirichlet's theorem on primes in arithmetic progression. However, deeper number theoretic results have to this day eluded this approach, for example, no one has succeeded in proving the prime number theorem by this method. Professor Riemann, aware of this deficiency, threw his notes into the wastebasket and proceeded to link the Riemann zeta function to the distribution of primes in an altogether different way, by stating the hypothesis that bears his name and that remains unproved to this day. \bigskip Why am I telling you this bit of history-fiction? Because I want to propose another probabilistic interpretation of the Riemann zeta function that is quite different from the interpretation just outlined. \bigskip Let us consider a problem in combinatorial enumeration. Let us take a cyclic group of order $r$, say $C_r $. Every character $\chi $ of the group $C_r$ has a kernel which is a subgroup of $C_r$. More generally, every sequence $\chi_1, \chi_2, \dots , \chi_s $ of characters of $C_r$ has a joint kernel which is also a subgroup of $C_r$; the joint kernel of a sequence of characters is simply the intersection of the kernels of each of the characters in the sequence $\chi_1, \chi_2, \dots , \chi_s $ . If a sequence $\chi_1, \chi_2, \dots , \chi_s $ of $s$ characters is chosen independently and at random, what is the probability that the joint kernel of the sequence equals a given subgroup $C_n $ of $C_r $? \bigskip The probability of the event that the kernel of a randomly chosen character will contain the subgroup $C_n$ equals $\frac{1}{n}$, since there are $r$ characters of the group $C_r$ and $\frac{r}{n}$ such characters will vanish on $C_n $. Therefore, the probability that the joint kernel of a randomly chosen sequence $\chi_1, \chi_2, \dots , \chi_s $ of $s $ characters shall contain the subgroup $C_n $ equals $(\frac{1}{n})^s $. Let us denote by $P_{C_n}$ the probability that the joint kernel of the characters $\chi_1, \chi_2, \dots , \chi_s $ shall equal the subgroup $C_n $. Then we have the identity $$ { 1 \over n^s} = \sum_{n|d|r} P_{C_d}.$$ Here, we use the fact that the partially ordered set of subgroups of a cyclic group $C_r$ is isomorphic to the partially ordered set of divisors of the integer $r$. \bigskip We now use the M\"obius inversion formula of number theory, thereby obtaining $$ P_{C_n} = \sum_{n|d|r} \mu (d/n){1 \over d^s}.$$ Here, $\mu(j) $ is the M\"obius function of number theory. \bigskip After the change of variable $d = nj $ we can recast the right hand side as follows: $$ P_{C_n} = { 1 \over n^s} \sum_{j}\mu (j){1 \over j^s}.$$ The variable $j$ on the right ranges over some subset of divisors of the integer $r$, which we need not worry about. \bigskip Now if the sum on the right ranged over all positive integers $j$, then the right hand side would equal $${ 1 \over n^s} \frac{1}{\zeta(s)},$$ \bigskip that is, it could be expressed in terms of the inverse of the Riemann zeta function. If we could change our combinatorial problem to get an unrestricted sum on the right hand side, then we would have a probabilistic interpretation of the Riemann zeta function. \bigskip This is done by replacing the finite cyclic group $C_n$ by a profinite cyclic group. \bigskip Consider the group $C_{\infty}$ of rational numbers modulo one. For every positive integer $n$, the group $C_{\infty}$ has a unique finite subgroup $C_n$ with $n$ elements. The character group $C_{\infty}^*$ of $C_{\infty}$ is a compact group; it has a Haar measure which is a probability measure $P$. The group $C_{\infty}^* $ is the desired profinite group on which we can generalize the preceding computation. \bigskip The set of all characters of the group $C_{\infty}$ (that is, the set of all elements of the group $C_{\infty}^* $) which vanish on a subgroup $C_n$ of $C_{\infty}$ has Haar measure equal to $\frac{1}{n}$. Thus, if we choose a sequence $\chi_1, \chi_2, \dots , \chi_s $ of $s $ characters of $C_{\infty}$ independently and at random, the probability that their joint kernel will contain the group $C_n$ equals $(\frac{1}{n})^s$ . If we again denote by $ P_{C_n} $ the probability that the joint kernel of a sequence $\chi_1, \chi_2, \dots , \chi_s $ of $s $ characters equals the group $C_n$, then we have the identity $${ 1 \over n^s} = \sum_{n|d} P_{C_d},$$ where the sum on the right is now infinite. Again by the M\"obius inversion formula we obtain $$ P_{C_n} = \sum_{n|d} \mu (d/n){ 1 \over d^s} = \frac{1}{n^s}\frac{1}{\zeta(s)}.$$ This is the promised probabilistic interpretation of the Riemann zeta function. Some properties of the Riemann zeta function can be proved probabilistically using this interpretation, for example, the product formula. It remains to be seen which other properties of the Riemann zeta function can be proved in this way. \bigskip The preceding argument is an instance of a generalization of an enumeration problem on a finite set to an enumeration on a profinite set. Such a replacement of a finite set by a profinite "set" works in other combinatorial problems. Will we ever have a profinite combinatorics on profinite sets side by side with combinatorics on finite sets? \newpage SECOND SNAPSHOT: THE CYCLIC DERIVATIVE. \bigskip The ordinary derivative of a polynomial in one variable has been generalized by Hausdorff to polynomials and formal power series in non commutative variables as follows. Consider the associative algebra ${\bf{C}} \langle a,b, \dots , c , x \rangle $ generated by a set of letters $\{a, b, \dots, c, x\}$. The letter $x$ is called a variable, all other letters are called constants. A monomial in this associative algebra is what you think it should be: it is a word like $$ m = axbax^3bcxd.$$ A polynomial is a linear combination of monomials, and a formal power series is defined as an infinite sum of monomials, with suitable restrictions on the growth of degrees of the summands. Formal power series in non commutative variables form an algebra ${\bf C}\langle \langle a,b,\dots , c, x \rangle \rangle $. We will denote by $f(x)$ such a formal power series. \bigskip The Hausdorff derivative of the monomial $m$ is computed as follows: $$ H (m) = H(axbax^3bcxd) = abax^3bcxd +3 axbax^2bcxd + axbax^3bcd. $$ This definition is extended by linearity to polynomials and to formal power series. \bigskip If $m'$ is another monomial, we have the expected rule for finding the Hausdorff derivative of a product: $H(m m') = H(m)m' + m H(m'). $ \bigskip The Hausdorff derivative suffers from a major weakness. There seems to be no analog of the chain rule for the differentiation of a function of a function. For example, the Hausdorff derivative of the polynomial $(ax)^n$, when the letters $a$ and $x $ do not commute, is not equal to $n (ax)^{n-1}a $. It is a mess. \bigskip There is another notion of derivative that does satisfy a simple chain rule under functional composition. It is the cyclic derivative, denoted by the letter $D$. \bigskip The cyclic derivative is defined as follows. First define the truncation operator $T$ as follows: \bigskip a. if the first letter of a monomial $m$ is not the variable $x$, set $T(m) = 0;$ \bigskip b. if the first letter of a monomial $m$ is the variable $x$, so that $m = x m' $, set $T(m) = m'. $ \bigskip c. Extend by linearity to ${\bf C}\langle \langle a,b,\dots , c, x \rangle \rangle $. \bigskip The cyclic derivative of a monomial $m$ is defined in terms of the truncation operator as follows: \bigskip a. Let $p$ be the polynomial obtained by adding all cyclic permutations of the monomial $m$. \bigskip b. Set $D(m) = T(p).$ \bigskip c. Extend by linearity to all formal power series. \bigskip For example, the cyclic derivative of the above monomial $m$ is computed in the following steps: \bigskip Step 1. Write down all cyclic permutations on the monomial $ axbax^3bcxd$. These are $$ xbax^3bcxda , bax^3bcxdax , ax^3bcxdaxb, x^3bcxdaxba , x^2bcxdaxbax , xbcxdaxbax^2 ,$$ $$bcxdaxbax^3 , cxdaxbax^3b , xdaxbax^3bc , daxbax^3bcx . $$ \bigskip Step 2. In the above list, perform one of the following operations: \bigskip a. if the first letter of a monomial is not $x$, remove the monomial from the list; \bigskip b. if the first letter of a monomial is $x$, remove the first letter. \bigskip When we perform operations a. and b. on the each of the monomials in the above list, we obtain a shorter list, namely: $$ bax^3bcxda , x^2bcxdaxba , xbcxdaxbax, bcxdaxbax^2 , daxbax^3bc. $$ Step 3. Add the monomials thus obtained to get the cyclic derivative: \begin{eqnarray*}D(m) & = & D(axbax^3bcxd) \\ & = & bax^3bcxda + x^2bcxdaxba + xbcxdaxbax+ bcxdaxbax^2 + daxbax^3bc. \end{eqnarray*} Another example: the cyclic derivative of the monomial $axbxcxdx$ equals $$D(axbxcxdx) = bxcxdxa + cxdxaxb + dxaxbxcx + axbxcxd. $$ The cyclic derivative of the monomial $(ax)^n$ is the following: \begin{eqnarray*} D((ax)^n) & = & D(axax \dots ax)\\ & = & axax\dots axa + axax\dots axa + \dots axax\dots axa \\ & = & n (ax)^{n-1} a.\end{eqnarray*} Similarly, one computes $$ D(x+a)^n = n(x+a)^{n-1}$$ and, for formal power series, $$ D(e^{x+a}) = e^{x+a} $$ and $$ D(e^{ax}) = e^{ax}a. $$ Remember, the letters $a$ and $x $ do not commute! In these examples, the corresponding Hausdorff derivative is a mess. The cyclic derivative enjoys all properties expected of the ordinary derivative; in particular, it satisfies the chain rule for the composition of two formal power series. \bigskip To state the rules for taking cyclic derivatives, we need one more operator, called the wrapping operator. \bigskip The wrapping operator is defined as follows. Let $c_1,c_2,\dots ,c_n $ be any letters. If $g(x)$ is any formal power series, set $$ \langle C c_1c_2\dots c_n | g(x) \rangle\\ = $$ $$c_1c_2\dots c_n g(x) + c_2\dots c_n g(x)c_1 + c_3 \dots c_n g(x)c_1c_2 + \dots + c_n g(x)c_1c_2\dots c_{n-1}.$$ \bigskip If $f(x)$ is any formal power series, the wrapping operator $$ \langle C f(x) | g(x) \rangle $$ is defined by linearity. \bigskip Define $$ \langle D(f(x)) | g(x) \rangle = T \langle C f(x) | g(x) \rangle. $$ For example: $$ \langle D(f(x)) | 1 \rangle = D(f(x)). $$ \bigskip The cyclic derivative of the product of two "functions" is given by the following identity: $$ D(f(x)g(x)) = \langle D(f(x)) | g(x) \rangle + \langle D(g(x)) | f(x) \rangle. $$ For example, one obtains \bigskip $$ D((1-ax)^{-1}(1-bx)^{-1}) = (1-ax)^{-1}(1-bx)^{-1}(1-ax)^{-1}a $$ $$ +(1-bx)^{-1}(1-ax)^{-1}(1-bx)^{-1}b .$$ \bigskip No such identity holds for the Hausdorff derivative. \bigskip The cyclic derivative of the product of any sequence of formal power series is similarly computed by the wrapping operator: $$D(f_1(x)f_2(x)\dots f_n(x)) = $$ $$\langle D(f_1(x)) | f_2(x) \dots f_n(x) \rangle + \langle D(f_2(x)) | f_3(x) \dots f_n(x)f_1(x) \rangle + $$ $$ \dots\\ + \langle D(f_n(x)) | f_1(x)f_2(x) \dots f_{n-1}(x) \rangle . $$ \bigskip We come now to the main property of the cyclic derivative: the chain rule. Given two formal power series $f(x) $ and $g(x) $ in ${\bf C}\langle \langle a,b,\dots , c, x \rangle \rangle $, assume that the formal power series $g(x) $ does not have a constant term. Under these circumstances, the composition $f(g(x))$ is well defined by replacing $g(x) $ for every occurrence of the variable $x$ in the formal power series $f(x)$. Let us write $ D_g(f(x)) $ to denote the formal power series obtained by substituting $g(x)$ in place of every occurrence of $x$ in the cyclic derivative $D(f(x))$ of the formal power series $f(x)$. Then the chain rule for the cyclic derivative goes as follows: $$ D(f(g(x))) = \langle Dg(x) | D_g(f(x)) \rangle. $$ For example, we have $$ D(e^{axbx}) = bx e^{axbx} a + e^{axbx}axb. $$ A more elegant example is the following: $$ D(e^{(1-ax)^{-1}}) = (1-ax)^{-1} e^{(1-ax)^{-1}}(1-ax)^{-1} a. $$ One can prove that the cyclic derivative of a rational formal power series in non commutative letters is again a rational non commutative power series, and that the cyclic derivative of an algebraic formal power series in non commutative letters is again an algebraic formal power series. \bigskip Despite the evidence that the cyclic derivative is the natural notion of derivative for non commutative algebras, the theory as it is at present is not satisfying. The cyclic derivative is an empirical discovery. It needs to be ensconced in some broader algebraic theory, much like the Hausdorff derivative has been ensconced in the theory of Hopf algebras. \newpage THIRD SNAPSHOT: LOGARITHMS AND THE BINOMIAL THEOREM. \bigskip The Euler-MacLaurin summation formula is one of the most remarkable formulas of mathematics. For a suitable function $f(x)$ of a real or complex variable, it is stated as follows: $$ f(x) + f(x+1) + f(x+2) + \dots + f(x+n) =$$ $$ B_0 \int_x^{x+n+1}f(y ) dy + B_1 (f(x+n+1) - f(x)) $$ $$ + {B_2 \over 2!} D (f(x+n+1) - f(x)) + {B_3 \over 3!} D^2 (f(x+n+1)- f(x)) + \dots .$$ The $B_n$ are the Bernoulli numbers and $D$ is the ordinary derivative operator. \bigskip The Euler-MacLaurin formula has proved very useful for over two hundred years. Nonetheless, the Euler-MacLaurin formula suffers from a serious deficiency. The series on the right hand side is almost never convergent, unless it reduces to a finite sum. \bigskip Our question is the following: is there a vector space of functions which contains as many of the elementary functions as possible, and a topology on such a vector space, relative to which the right hand side of the Euler-MacLaurin formula is a convergent series? \bigskip The answer to this question is unexpectedly related to the answer to another question. What is the "right" generalization of the binomial coefficients ${n \choose k}$ when $k$ is allowed to be a negative integer? This question leads in turn to a third question: how shall we know whether a generalization of the binomial coefficients is "right"? The answer to this third question is easy: a generalization of the binomial coefficients is "right" if it leads to a sensible generalization of the binomial theorem: \bigskip $$ (a+ x)^n = \sum_{k=0}^n {n \choose k}a^k x^{n-k}. $$ \bigskip When I was young, I used to think of the binomial theorem as trivial. I think I have learned my lesson. A well-known philosopher, I can't remember his name, wrote that the whole universe can be inferred from a grain of sand. He should have added that a great deal of mathematics can be derived by meditating upon the binomial theorem. \bigskip Let us take the bull by the horns, and state the "right" generalization of the binomial coefficients. We proceed in the most pedestrian way, by first generalizing the definition of the factorial. Thus, let $n$ be any integer, positive or negative. We define the Roman factorial $[n]!$ as follows: $$ [n]! = n! $$ if $ n \geq 0 $, and $$ [n]! = {(-1)^{n+1} \over (-n-1)!} $$ if $ n < 0 $. \bigskip Where does this definition come from ? I could simply say that it works, but that would not be the whole truth. The value of the Roman factorial $ [n]! $ for $n$ negative equals the residue of the gamma function at the integer $n-1$. Using the Roman factorial, we define the Roman coefficients as follows: $$ { n \brack k} = { [n]! \over [k]![n-k]!}. $$ When $ n \geq k \geq 0 $, the Roman coefficients coincide with the binomial coefficients. For all integers $n$ and $k$, the Roman coefficients share all elementary properties of binomial coefficients, such as Pascal's triangle, etc. However, there are some surprises in store, for example, for $k$ positive we find $$ {0 \brack -k}= {0 \brack k} = { (-1)^{k+1} \over k }. $$ \bigskip Does this make any sense? Well, yes, because we can find a generalization of the binomial theorem that goes with this. It is the following. Recall the power series expansion of the logarithm: $$ log(x+a) = log x + \sum_{k=1}^{\infty}{ (-1)^{k+1} \over k} {a^k \over x^k}.$$ We can recast this power series expansion in terms of the Roman coefficients as follows: $$ log(x+a) = log x + \sum_{k=1}^{\infty} {0 \brack k} {a^k \over x^k}. $$ This is beginning to look like a generalization of the binomial theorem, with the logarithm playing the roles of zero-th power. Another power series expansion where the Roman coefficients make their appearance is the following: $$ (x+a)(log(x+a) - 1) = x(log x - 1) + a log x + \sum_{k=2}^{\infty}{1 \brack k} a^k x^{1 -k}. $$ Do we see a pattern? Well, let us try yet another power series expansion: $$ (x +a)^2(log(x+a) - 1 - {1 \over 2}) =$$ $$ x(log x - 1 - {1 \over 2}) + {2 \brack 1}a x(log x - 1) + {2 \brack 2}a^2 log x + \sum_{k=3}^{\infty} {2 \brack k} a^k x^{2-k}. $$ Now we can leap to a generalization. For suitable functions $f(x)$, set $$ D^{-1}f(x) $$ to be the unique indefinite integral of the function $f(x)$ which has constant term equal to zero. Do not worry, this will make sense in a moment. \bigskip Define $$ \lambda_n^{(1)} (x) = [n]! D^{-n}log x. $$ Here, $n$ is any integer, positive or negative. The functions $\lambda_n^{(1)}(x)$ are called the harmonic logarithms of order one. For $n$ positive we have $$ \lambda_n^{(1)} (x) = x^n(log x - 1 - {1\over 2} - {1 \over 3} - \dots -{1 \over n}) $$ and $$ \lambda_{-n}^{(1)} (x) = {1 \over x^n}.$$ Of course we also have $$ \lambda_0^{(1)} (x) = log x.$$ We are now in a position to state the generalization of the binomial theorem that is associated with the harmonic logarithms. It goes as follows: $$ \lambda_n^{(1)} (x+ a) = \sum_{k=0}^{\infty}{n \brack k} a^k \lambda_{n-k}^{(1)} (x). $$ The three identities above are special cases of this identity, for $n = 0,1,2$. \bigskip The generalization of the binomial theorem to harmonic logarithms gives nothing new for negative exponents, where it reduces to the identity $$(x+a)^{-n} = \sum_{k=0}^{\infty} { -n \choose k} a^k x^{-n-k}.$$ However, for positive exponents we obtain a genuine and baffling generalization of the binomial theorem. It states that the functions $ \lambda_n^{(1)} (x) $, for $n $ positive, satisfy the ordinary binomial theorem, modulo negative powers of $x$. In other words, we have the following identity: $$ (x+a)^n(log(x+a) - 1 - {1\over 2} - {1 \over 3} - \dots - {1 \over n}) \cong $$ $$\sum_{k=0}^n {n \choose k}a^k x^{n-k}(log(x) - 1 - {1\over 2} - {1 \over 3} - \dots - {1 \over {n-k}}). $$ The identity is valid modulo negative powers of $x$. Miracles of cancellation are occurring in this identity. I wish I knew a combinatorial or probabilistic interpretation of this logarithmic generalization of the binomial theorem. \bigskip So far, we have assumed that all series converge in the topology of the complex numbers. We will now change the topology, while retaining the convergence. \bigskip The motivation for the logarithmic topology we are about to define is the algebra of formal Laurent series. This topological algebra may be defined by defining a topology on the algebra of rational functions in the variable $x$, and then completing this algebra relative to the topology. The topology is so chosen as to have $lim_{n \rightarrow \infty}x^{-n} = 0 $. Every element of the completed algebra turns out to be a formal Laurent series, that is, a series of the form $$ \sum_{n < d} a_n x^n. $$ We want to perform an analogous completion process on another algebra: the algebra generated by all functions of the form $x^n(logx)^t$, where $n$ is any integer, positive or negative, and where $t$ is a non negative integer. In order to specify which elements of this algebra are to converge to zero, we need a better behaved basis of this algebra. This basis is provided by the harmonic logarithms of arbitrary order $t$. They are defined as follows: $$\lambda_n^{(t)}(x) =[n]! D^n (log x)^t $$ for every nonnegative integer $t$ and for every integer $n$ . For example, we have $$\lambda_n^{(0)}(x) = x^n $$ for every nonnegative integer $n$, and $$ \lambda_n^{(0)}(x) = 0 $$ for negative $n$. \bigskip Explicit expressions are known for the harmonic logarithms. For the harmonic logarithms of order $2$ we have $\lambda_0^{(2)}(x) = (logx)^2 $ and for $n$ positive $$\lambda_n^{(2)}(x) = x^n \big( (logx)^2 - ( 2 + {2 \over 2} + \dots + {2 \over n})log x + 2 + {2 \over 2} (1+ {1 \over 2}) + \dots + {2 \over n}(1+ {1 \over 2} + \dots + {1 \over n}) \big), $$ and $$ \lambda_{-n}^{(2)}(x) = 2x^{-n} \big( log x - 1 - {1\over 2} - \dots - {1 \over n-1} \big). $$ For every non negative integer $t$ the harmonic logarithms of order $t$ satisfy the same generalization of the binomial theorem that we have already seen for the harmonic logarithms of order $1$: $$ \lambda_n^{(t)} (x+ a) = \sum_{k=0}^{\infty}{n \brack k} a^k \lambda_{n-k}^{(t)} (x). $$ The harmonic logarithms are a basis of the algebra generated by all functions $x^n(logx)^t$. We define a topology on this algebra by requiring that $$ lim_{n \rightarrow - \infty}\lambda_n^{(t)}(x) = 0 $$ for every non negative integer $t$. This topology is called the logarithmic topology. The completion of this algebra relative to the logarithmic topology is the algebra of formal power series of logarithmic type, or logarithmic algebra. \bigskip Every element of the logarithmic algebra is a linear combination of convergent power series of the form $$ f(x) = \sum_{t, n \leq d} b_{n,t} \lambda_n^{(t)}(x) $$ ranging over a finite set of values of $t$. \bigskip We can now return to the Euler-MacLaurin summation formula: \bigskip {\bf Theorem}. For every element $f(x)$ of the logarithmic algebra the right hand side of the Euler-MacLaurin series converges in the logarithmic topology. \bigskip For example, the following infinite series is convergent in the logarithmic topology: $$ log x + log(x+1) + log(x+2) + \dots + log(x+n) = $$ $$B_0 ((x+n+1)log(x+n+1) - xlogx - n - 1 ) + B_1 ( log(x+n+1) - logx ) +$$ $${B_2 \over 2!} ({1 \over x+n+1 } - {1 \over x}) + \dots ) . $$ \bigskip Another example is the following. As you know, the sum $$ x^k + (x+1)^k + (x+2)^k + \dots + (x+n)^k $$ can be expressed in closed form by the Euler-Maclaurin formula. The preceding theorem leads to analogous closed form expressions for sums of the form $$ x^klogx + (x+1)^klog(x+1) + (x+2)^klog(x+2)+ \dots + (x+n)^klog(x+n) .$$ The harmonic logarithms have other applications, let us mention one in closing. \bigskip Recall the definition of the shift operator of the calculus of finite differences: $$E^af(x) = f(x+a).$$ For $n$ a non negative integer, define the operator $E_1$ as follows: $$ E_1 \lambda_n^{(0)}(x) = \lambda_n^{(1)}(x). $$ In ordinary notation, this is the same as saying $$ E_1 x^n = x^n(log x - 1 - 1/2 - 1/3 - \dots - 1/n). $$ One can prove the following two propositions: \bigskip {\bf Proposition}. The operators $E^a$ and $E_1$ commute. \bigskip {\bf Proposition}. The restriction of the derivative operator $D$ to the subalgebra of the logarithmic algebra generated by the harmonic logarithms $\lambda_n^{(t)}(x)$ for positive $t$ (that is, excluding the non negative powers of $x$) is invertible. \bigskip These two propositions can be used to obtain "logarithmic extensions" of special functions. Let us conclude with the simplest example: let us compute the logarithmic extension of the sequence of lower factorials, namely, the polynomials $ (x)_n = x(x-1)(x-2)\dots (x-n+1)$. This sequence of polynomials satisfies the difference equation $$ \Delta (x)_n = n(x)_{n-1},$$ where $\Delta$ is the difference operator: $\Delta f(x) = f(x+1) - f(x).$ This sequence can be extended to negative $n$ by setting $$ (x)_{-n} = { 1 \over {(x+1)(x+2)...(x+n)} }, $$ and we have $$ \Delta (x)_{-n} = -n(x)_{-n-1}.$$ For positive $n$, we may define the logarithmic extension of this sequence by setting $$ (x)_{-n}^{(1)} = (x)_{-n} ={ 1 \over {(x+1)(x+2)...(x+n)} }. $$ For example, $(x)_{-1}^{(1)} = { 1\over x+1}$. \bigskip The elements $ (x)_{-n}^{(1)}$ belong to the submodule of the logarithmic algebra spanned by $\lambda_n^{(1)}(x)$, as $n $ ranges over all integers. On this submodule, the operator $\Delta$ is invertible, and we can therefore set $$ (x)_n^{(1)} = \Delta^{-n-1}{1 \over x+1}$$ for all non negative integers $n$. \bigskip It turns out that the element $(x)_0^{(1)}$ is given by the following series, convergent in the logarithmic topology: $$ (x)_0^{(1)} = log(x+1) + {B_1 \over 1+x} - {B_2 \over 2(1+x)^2} + {B_3 \over { 3(1+x)^3 }} - \dots . $$ But this is a familiar object: it is the $\Psi$- function, heuristically introduced by Gauss. Gauss motivated the $\Psi$-function as the "right" solution of the difference equation $$ \Delta \Psi(x+1) = {1 \over { x+1}}. $$ We have now rigorously verified Gauss's guess. Further computations show that the elements $ (x)_1^{(1)}$ and $ (x)_2^{(1)} $ also coincide with special functions introduced by Gauss, namely, the digamma and trigamma functions, which are at last rigorously defined by infinite series convergent in the logarithmic topology. \bigskip In a similar vein, one defines logarithmic extensions of the Bernoulli polynomials, the Hermite polynomials, etc., and one finds that the asymptotic expansions of these polynomials reappear naturally as members of the logarithmic extensions of these functions. As a matter of fact, the logarithmic topology allows us to replace asymptotic expansions by series which are convergent in the logarithmic topology. \bigskip In closing, two open problems may be mentioned. \bigskip First, no closed form expression is known for the coefficients of the expansion of a product $$\lambda_n^{(t)}(x) \lambda_k^{(s)}(x) $$ into a logarithmic power series. Second, we do not know a combinatorial or probabilistic interpretation of the Roman coefficients ${n \brack k}$ in general. \bigskip Thank you for listening. \bigskip {\bf Bibliography} \bigskip J.P. S. Kung, M. Ram Murthy and Gian-Carlo Rota, The Redei zeta function, Journal of Number Theory, vol. 12 (1980), pp. 421-436. \bigskip Gian-Carlo Rota, Bruce Sagan and Paul R. Stein, A cyclic derivative in noncommutative algebra, Journal of Algebra, vol. 64 (1980), pp.54-75/ \bigskip Daniel E. Loeb and Gian-Carlo Rota, Formal power series of logarithmic type, Advances in Mathematics, vol.75 (1989), pp. 1-118. \bigskip Kenneth S. Alexander, Kenneth Baclawski and Gian-Carlo Rota, A stochastic interpretation of the Riemann zeta function, Proceedings of the National Academy of Sciences, vo. 90 (1993), pp. 697-699. \bigskip J.P.S. Kung, ed., Gian-Carlo Rota on Combinatorics, Birkh\"auser Boston, 1996, ISBN 0-8176-3713-3 or ISBN 3-7643-3713-3 \bigskip \end{document} --============_-1322120415==_============-- From sara(at-sign)math.mit.edu Wed Mar 18 14:45:11 1998 Return-Path: Received: from schubert.mit.edu (SCHUBERT.MIT.EDU [18.87.0.16]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id OAA17979 for ; Wed, 18 Mar 1998 14:45:11 -0500 (EST) Received: (from sara(at-sign)localhost) by schubert.mit.edu (8.8.7/8.8.7) id OAA20244; Wed, 18 Mar 1998 14:45:10 -0500 (EST) Date: Wed, 18 Mar 1998 14:45:10 -0500 (EST) Message-Id: <199803181945.OAA20244(at-sign)schubert.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara Note, the seminar is basically booked through the end of the semester If anyone didn't get a chance to speak who wanted to, please let me know so I can schedule a talk in the fall. *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Speaker: Alex Suciu Title: Characteristic varieties of real and complex arrangements Abstract: The k-th Fitting ideal of the Alexander invariant of an arrangement A of n complex hyperplanes defines a characteristic subvariety, V_k(A), of the complex algebraic torus (C^*)^n. The characteristic varieties of an arrangement provide subtle and effectively computable homotopy-type invariants of its complement. In joint work with Daniel Cohen, we show that the tangent cone at the identity of the top characteristic variety V_1(A) coincides with R_1(A), the first-cohomology support locus of the Orlik-Solomon algebra. We conclude that the variety V_1(A) is combinatorially determined, and that Falk's variety R_1(A) is the union of a subspace arrangement in C^n. We illustrate these techniques by computing the top characteristic varieties of braid arrangements and monomial arrangements. If A is a real 2-arrangement (in the sense of Goresky and McPherson), the characteristic varieties are no longer subtori through the origin. The nature of these varieties vividly illustrates the difference between real and complex arrangements. In joint work with Daniel Matei, we study the homotopy types of complements of arrangements of n transverse planes in R^4, obtaining a complete classification for n<=6, and lower bounds for the number of homotopy types in general. Furthermore, we show that the homotopy type of the complement of a 2-arrangement in R^4 is not determined by its cohomology ring, thereby answering a question of Ziegler. The papers on which the talk will be based can be found at http://www.math.neu.edu/~suciu/publications.html. April 1: Jim Haglund Further investigations involving polynomials with only real roots April 8: Irena Peeva Monomial ideals, real Boolean subspace arrangements, and order dimension of lattices April 10: Karen Collins Breaking symmetries of S5 and S6 April 17: Anders Buch Chern class formulas for degeneracy loci April 22: Andrzej Rucinski On graphs with linear Ramsey numbers April 24: Lou Shapiro Path pairs and Catalan probabilities April 29: Patricia Hersh Shuffle posets of multisets ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From kcollins(at-sign)mail.wesleyan.edu Wed Mar 25 10:15:42 1998 Return-Path: Received: from mail.wesleyan.edu (dns.wesleyan.edu [129.133.12.10]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id KAA13286; Wed, 25 Mar 1998 10:15:41 -0500 (EST) Received: from [129.133.30.205] (kcollins1.math.wesleyan.edu [129.133.30.205]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id KAA12215; Wed, 25 Mar 1998 10:13:49 -0500 (EST) X-Sender: kcollins(at-sign)mail.wesleyan.edu Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 25 Mar 1998 10:18:30 -0500 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: second announcement Come to the Twenty-ninth one day conference on Combinatorics and Graph Theory Saturday, April 4, 1998 10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063 Schedule 10:00 Van Vu (Yale University) Witness on the Upper Bound of Chromatic (Choice) Number of Random Graphs 11:10 Richard Stanley (MIT) Spanning Trees and a Conjecture of Kontsevich 12:10 Lunch 2:00 Eckhard Steffen (Princeton University) Snarks 3:10 Therese Biedl (McGill University) Efficient Algorithms For Petersen's Theorem The next meeting is scheduled for May 2nd. The conferences are supported by an NSF grant which allows us to provide a modest transportation allowance to those attendees who are not local. We also gratefully acknowledge support from Smith College and Wesleyan University. Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)math.smith.edu Karen Collins (Wesleyan University), (860) 685-2169, kcollins(at-sign)wesleyan.edu Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)math.smith.edu From sara(at-sign)math.mit.edu Fri Mar 27 15:06:47 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id PAA05631; Fri, 27 Mar 1998 15:06:47 -0500 (EST) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id PAA12142; Fri, 27 Mar 1998 15:06:47 -0500 (EST) Date: Fri, 27 Mar 1998 15:06:47 -0500 (EST) Message-Id: <199803272006.PAA12142(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminars in April X-URL: http://www-math.mit.edu/~sara Hope you are all enjoying spring break. No talk today. Below is the schedule for April. I just added a second talk this coming week. All abstracts are on the web as usual: http://www-math.mit.edu/~combin April 1: Jim Haglund Further investigations involving polynomials with only real roots April 3: Manjul Bhargava The factorial function... and generalizations April 8: Irena Peeva Monomial ideals, real Boolean subspace arrangements, and order dimension of lattices April 10: Karen Collins Breaking symmetries of S5 and S6 April 15: Patricia Hersh Shuffle posets of multisets April 17: Anders Buch Chern class formulas for degeneracy loci April 22: Andrzej Rucinski On graphs with linear Ramsey numbers April 24: Lou Shapiro Path pairs and Catalan probabilities April 29: To be announced. ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From propp(at-sign)math.mit.edu Mon Mar 30 12:12:10 1998 Return-Path: Received: from pfaff.mit.edu (PFAFF.MIT.EDU [18.87.0.183]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id MAA14300 for ; Mon, 30 Mar 1998 12:12:10 -0500 (EST) Received: (from propp(at-sign)localhost) by pfaff.mit.edu (8.8.7/8.6.9) id MAA19141 for combinatorics(at-sign)math.mit.edu; Mon, 30 Mar 1998 12:12:05 -0500 (EST) Date: Mon, 30 Mar 1998 12:12:05 -0500 (EST) From: Jim Propp Message-Id: <199803301712.MAA19141(at-sign)pfaff.mit.edu> To: combinatorics(at-sign)math.mit.edu Subject: test message --- please ignore I'm just checking to make sure that auto-archiving is once again working properly. Jim Propp From kcollins(at-sign)mail.wesleyan.edu Tue Mar 31 13:03:24 1998 Return-Path: Received: from mail.wesleyan.edu (dns.wesleyan.edu [129.133.12.10]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA13522; Tue, 31 Mar 1998 13:03:24 -0500 (EST) Received: from [129.133.30.205] (kcollins1.math.wesleyan.edu [129.133.30.205]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id NAA23909; Tue, 31 Mar 1998 13:01:18 -0500 (EST) X-Sender: kcollins(at-sign)mail.wesleyan.edu Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Tue, 31 Mar 1998 13:07:13 -0500 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: third announcement Come to the Twenty-ninth one day conference on Combinatorics and Graph Theory Saturday, April 4, 1998 10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063 Schedule 10:00 Van Vu (Yale University) Witness on the Upper Bound of Chromatic (Choice) Number of Random Graphs 11:10 Richard Stanley (MIT) Spanning Trees and a Conjecture of Kontsevich 12:10 Lunch 2:00 Eckhard Steffen (Princeton University) Snarks 3:10 Therese Biedl (McGill University) Efficient Algorithms For Petersen's Theorem The next meeting is scheduled for May 2nd. The conferences are supported by an NSF grant which allows us to provide a modest transportation allowance to those attendees who are not local. We also gratefully acknowledge support from Smith College and Wesleyan University. Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)math.smith.edu Karen Collins (Wesleyan University), (860) 685-2169, kcollins(at-sign)wesleyan.edu Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)math.smith.edu From sara(at-sign)math.mit.edu Tue Mar 31 14:14:23 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id OAA15732 for ; Tue, 31 Mar 1998 14:14:23 -0500 (EST) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id OAA27880; Tue, 31 Mar 1998 14:14:23 -0500 (EST) Date: Tue, 31 Mar 1998 14:14:23 -0500 (EST) Message-Id: <199803311914.OAA27880(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar tomorrow X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar TOMORROW at 4:15 in 2-338********** refreshments served at 3:45 ** Pretalk at 3:30-4:00** Speaker: Jim Haglund (MIT) Title: Further investigations involving polynomials with only real roots Preprint: http://www-math.mit.edu/~combin/preprints Abstract: We present a number of conjectures involving rook polynomials having only real zeros. Many of these generalize a previous conjecture of the author, K. Ono, and D. G. Wagner, namely that if $A$ is a real $n \times n$ matrix which is weakly increasing down columns, then the permanent of $zA+J_n$ has only real zeros. In some cases these conjectures are motivated by factorization theorems for Ferrers boards. Connections between results of Laguerre and Szeg\"o on transformations which send polynomials with only real roots to other such polynomials are discussed. We also present a weighted version of the Poset Conjecture of enumerative combinatorics. Upcoming Events: April 3: Manjul Bhargava The factorial function... and generalizations April 8: Irena Peeva Monomial ideals, real Boolean subspace arrangements, and order dimension of lattices April 10: Karen Collins Breaking symmetries of S5 and S6 April 15: Patricia Hersh Shuffle posets of multisets April 17: Anders Buch Chern class formulas for degeneracy loci April 22: Andrzej Rucinski On graphs with linear Ramsey numbers April 24: Lou Shapiro Path pairs and Catalan probabilities ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From bergeron(at-sign)pascal.math.yorku.ca Wed Apr 1 16:32:09 1998 Return-Path: Received: from sungod.ccs.yorku.ca (syMqAHmp+f0PWXF+Yvf4CINiPsmIm18q(at-sign)[130.63.236.104]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id QAA17677 for ; Wed, 1 Apr 1998 16:32:05 -0500 (EST) Received: from [130.63.218.86] (msfac1.math.yorku.ca [130.63.218.86]) by sungod.ccs.yorku.ca (8.8.7/8.6.11) with SMTP id QAA09873; Wed, 1 Apr 1998 16:32:09 -0500 (EST) X-Sender: bergeron(at-sign)pascal.math.yorku.ca Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 1 Apr 1998 16:39:32 -0400 To: combopt(at-sign)math.uwaterloo.ca (Combinatorics and Optimization Waterloo), combinatorics(at-sign)math.mit.edu, combinatorics(at-sign)euclid.UCSD.EDU, combinatorics(at-sign)cs.toronto.edu, blaw(at-sign)fields.toronto.edu From: bergeron(at-sign)pascal.math.yorku.ca (Nantel Bergeron) Subject: FPSAC/SFCA 98 ------------------------------------------------------------------------- 10-th Conference on Formal Power Series and Algebraic Combinatorics 10ieme conference sur les Series Formelles et la Combinatoire Algebriques June 14 - 19 juin, 1998 Fields Institute, Toronto ------------------------------------------------------------------------- OnLine Registration: reduced registration fees... before April 15 Inscription en ligne: frais reduit... avant le 15 avril "http://www.math.yorku.ca/bergeron" ------------------------------------------------------------------------- Full Schudule and other informations: Horraire complet et autres informations: "http://www.math.yorku.ca/bergeron" ------------------------------------------------------------------------- INVITED TALKS/CONFERENCIER INVITES: G. Benkart (USA) P. Cameron (England) P. Dehornoy (France) B. Derrida (France) P. Diaconis (USA) C. Godsil (Canada) K. Ono (USA) J. Y. Thibon (France) B. Sturmfels (USA) ------------------------------------------------------------------------- For more information contact: Pour plus de renseignements contactez: FPSAC/SFCA 98 Fields Institute 222 College Street Toronto, ON M5T 3J1 Canada fpsac98(at-sign)fields.utoronto.ca bergeron(at-sign)mathstat.yorku.ca. ------------------------------------------------------------------------- ORGANIZATION/COMITE D'ORGANIZATION: N. Bergeron, Chairman (York U.), M. Delest (U. de Bordeaux), F. Sottile (U. Toronto), W. W hiteley (York U.). ------------------------------------------------------------------------- Nantel Bergeron bergeron(at-sign)mathstat.yorku.ca Associate Prof. Mathematics nantel(at-sign)math.harvard.edu York University nantel(at-sign)lacim.uqam.ca http://www.math.yorku.ca/bergeron From sara(at-sign)math.mit.edu Fri Apr 3 12:01:57 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id MAA01648 for ; Fri, 3 Apr 1998 12:01:57 -0500 (EST) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id MAA17543; Fri, 3 Apr 1998 12:01:56 -0500 (EST) Date: Fri, 3 Apr 1998 12:01:56 -0500 (EST) Message-Id: <199804031701.MAA17543(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Speaker: Manjul Bhargava Title: The factorial function... and generalizations Abstract: Though ubiquitous in combinatorics, the factorial function also occurs surprisingly often in number theory. A close examination of these occurrences leads to a series of beautiful generalizations of the factorial function, which also turn out to be quite useful in a variety of combinatorial, number-theoretic, and ring-theoretic problems. In particular, a fundamental problem about integer-valued polynomials, put forth by P\'olya in 1919, is now resolved. Upcoming events: April 8: Irena Peeva Monomial ideals, real Boolean subspace arrangements, and order dimension of lattices April 10: Karen Collins Breaking symmetries of S5 and S6 April 15: Patricia Hersh Shuffle posets of multisets April 17: Anders Buch Chern class formulas for degeneracy loci April 22: Andrzej Rucinski On graphs with linear Ramsey numbers April 24: Lou Shapiro Path pairs and Catalan probabilities ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From sara(at-sign)math.mit.edu Wed Apr 8 13:25:21 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA23722 for ; Wed, 8 Apr 1998 13:25:21 -0400 (EDT) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id NAA26827; Wed, 8 Apr 1998 13:25:20 -0400 (EDT) Date: Wed, 8 Apr 1998 13:25:20 -0400 (EDT) Message-Id: <199804081725.NAA26827(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Speaker: Irena Peeva (MIT) Title: Monomial ideals, real Boolean subspace arrangements, and order dimension of lattices Abstract: The talk is on a joint work with V. Gasharov, V. Reiner, and V. Welker. We give a formula for the graded Betti numbers of a monomial ideal in terms of the homology of the lower intervals in the lattice of least common multiples of the minimal monomial ideal generators; the formula has the same flavour as the Goresky-MacPherson formula for the dimensions of the cohomology of the complement of a subspace arrangement. This leads to a relation of the cohomological properties of real Boolean arrangements and square-free monomial ideals. Another application of the formula is a relation between the homology and the order dimension of an arbitrary finite lattice. Upcoming Events: April 10: Karen Collins Breaking symmetries of S5 and S6 April 15: Patricia Hersh Shuffle posets of multisets April 17: Anders Buch Chern class formulas for degeneracy loci April 22: Andrzej Rucinski On graphs with linear Ramsey numbers April 24: Lou Shapiro Path pairs and Catalan probabilities ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From sara(at-sign)math.mit.edu Fri Apr 10 12:50:36 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id MAA17783 for ; Fri, 10 Apr 1998 12:50:36 -0400 (EDT) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id MAA00765; Fri, 10 Apr 1998 12:50:01 -0400 (EDT) Date: Fri, 10 Apr 1998 12:50:01 -0400 (EDT) Message-Id: <199804101650.MAA00765(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Speaker: Karen Collins Title: Breaking symmetries of S5 and S6 Abstract: A labeling of the vertices of a graph G is said to be r-distinguishing, provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, D(G), is the minimum r such that G has an r-distinguishing labeling. For any group J, we define D(J) := { D(G) | Aut(G) is isomorphic to J }. It has been shown that D(S3) = {2,3}, and D(S4) = {2,4}. In this talk, we will sketch the proof that D(S5) = {2,3,5} and present other distinguishing results. Upcoming Events: April 15: Patricia Hersh Shuffle posets of multisets April 17: Anders Buch Chern class formulas for degeneracy loci April 22: Andrzej Rucinski On graphs with linear Ramsey numbers April 24: Lou Shapiro Path pairs and Catalan probabilities ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From kcollins(at-sign)mail.wesleyan.edu Sun Apr 12 10:09:58 1998 Return-Path: Received: from mail.wesleyan.edu (dns.wesleyan.edu [129.133.12.10]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id KAA21997; Sun, 12 Apr 1998 10:09:57 -0400 (EDT) Received: from [129.133.30.205] (kcollins1.math.wesleyan.edu [129.133.30.205]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id KAA13815; Sun, 12 Apr 1998 10:09:33 -0400 (EDT) X-Sender: kcollins(at-sign)mail.wesleyan.edu Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Sun, 12 Apr 1998 10:14:18 -0400 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: May 2nd Come to the Thirtieth one day conference on Combinatorics and Graph Theory Saturday, May 2, 1998 10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063 Schedule 10:00 Dan Rockmore (Dartmouth College) FFTs for $SL_2(F_q)$ - Theory and Applications 11:10 Randall McCutcheon (Wesleyan Univ.) TBA 12:10 Lunch 2:00 Joel Spencer (New York Univ. -- Courant Institute) Sixty Years of Ramsey R(3,k) 3:10 Lakshmibai (Northeastern Univ.) Flag varieties -- their Geometry & Combinatorics The conferences are supported by an NSF grant which allows us to provide a modest transportation allowance to those attendees who are not local. We also gratefully acknowledge support from Smith College and Wesleyan University. Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)math.smith.edu Karen Collins (Wesleyan University), (860) 685-2169, kcollins(at-sign)wesleyan.edu Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)math.smith.edu From propp(at-sign)math.mit.edu Wed Apr 15 13:57:18 1998 Return-Path: Received: from pfaff.mit.edu (PFAFF.MIT.EDU [18.87.0.183]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA12003 for ; Wed, 15 Apr 1998 13:57:18 -0400 (EDT) Received: (from propp(at-sign)localhost) by pfaff.mit.edu (8.8.7/8.6.9) id NAA13816 for combinatorics(at-sign)math.mit.edu; Wed, 15 Apr 1998 13:57:18 -0400 (EDT) Date: Wed, 15 Apr 1998 13:57:18 -0400 (EDT) From: Jim Propp Message-Id: <199804151757.NAA13816(at-sign)pfaff.mit.edu> To: combinatorics(at-sign)math.mit.edu Subject: Discrete Dinner After a two-year hiatus, the BOSTON AREA DISCRETE DINNER is back. The proposed dates for this term are May 6, May 8, and May 13. For each of the proposed dates, please let me know what your probability of attendance would be, if the dinner were held on that day. If you have any particular recommendations for local restaurants, feel free to submit those as well. Also feel free to pass this message on to combinatorialists and discrete mathematicians who aren't on the recipient list. Jim Propp (propp(at-sign)math.mit.edu) From sara(at-sign)math.mit.edu Wed Apr 15 15:00:51 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id PAA14205 for ; Wed, 15 Apr 1998 15:00:51 -0400 (EDT) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id PAA12250; Wed, 15 Apr 1998 15:00:51 -0400 (EDT) Date: Wed, 15 Apr 1998 15:00:51 -0400 (EDT) Message-Id: <199804151900.PAA12250(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Speaker: Patricia Hersh Title: Shuffle posets of multisets Abstract: We study posets defined by Stanley as a multiset generalization of Greene's shuffle posets. We compute the flag f-vector in terms of the quasi-symmetric function encoding $F_P$ defined by Ehrenborg. The result is a symmetric function which is expressible as a sum of Schur functions with nonnegative integer coefficients. We find a topological chain decomposition which yields $F_P$ and similarly gives the characteristic polynomial, zeta polynomial and rank generating function for shuffle posets of multisets. This decomposition also leads to a symmetric group action on maximal chains with Frobenius characteristic related to $F_P$. Upcoming Events: April 17: Anders Buch Chern class formulas for degeneracy loci April 22: Andrzej Rucinski On graphs with linear Ramsey numbers April 24: Lou Shapiro Path pairs and Catalan probabilities **April 29: Henrik Eriksson Sorting bridge hands and DNA. ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From sara(at-sign)math.mit.edu Fri Apr 17 13:57:36 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA07498 for ; Fri, 17 Apr 1998 13:57:36 -0400 (EDT) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id NAA15207; Fri, 17 Apr 1998 13:57:36 -0400 (EDT) Date: Fri, 17 Apr 1998 13:57:36 -0400 (EDT) Message-Id: <199804171757.NAA15207(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Speaker: Anders Buch Title: Chern class formulas for degeneracy loci Preprint:"http://www-math.mit.edu/~combin/abstracts/apr98/quiver.ps" Abstract: We study a general class of degeneracy loci associated to a sequence of vector bundles with maps between them and arbitrary rank conditions on the maps and their compositions. The cohomology classes of such loci are described by polynomials in the Chern classes of the vector bundles. These polynomials generalize all known types of Schubert polynomials. We give explicit formulas for the polynomials as linear combinations with integer coefficients of products of Schur determinants. We furthermore conjecture that all coefficients are positive and given by counting tableaux. Upcoming Events: April 22: Andrzej Rucinski On graphs with linear Ramsey numbers April 24: Lou Shapiro Path pairs and Catalan probabilities April 29: Henrik Eriksson. Sorting bridge hands and DNA May 6:David Ingerman (NYU) Inverse boundary problems, arrangements of pseudo-lines and total positivity May 8: Christian Lenart (MIT) Combinatorial Aspects of the K-theory of flag varieties May 13: Jim Propp (MIT) Diabolo tilings of fortresses ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From kcollins(at-sign)mail.wesleyan.edu Mon Apr 20 10:12:41 1998 Return-Path: Received: from mail.wesleyan.edu (dns.wesleyan.edu [129.133.12.10]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id KAA19267; Mon, 20 Apr 1998 10:12:40 -0400 (EDT) Received: from [129.133.30.205] (kcollins1.math.wesleyan.edu [129.133.30.205]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id KAA24427; Mon, 20 Apr 1998 10:11:11 -0400 (EDT) X-Sender: kcollins(at-sign)mail.wesleyan.edu Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Mon, 20 Apr 1998 10:16:15 -0400 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: second announcement Come to the Thirtieth one day conference on Combinatorics and Graph Theory Saturday, May 2, 1998 10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063 Schedule 10:00 Dan Rockmore (Dartmouth College) FFTs for $SL_2(F_q)$ - Theory and Applications 11:10 Randall McCutcheon (Wesleyan Univ.) Sets of recurrence in semigroups and Ramsey theory 12:10 Lunch 2:00 Joel Spencer (New York Univ. -- Courant Institute) Sixty Years of Ramsey R(3,k) 3:10 Lakshmibai (Northeastern Univ.) Flag varieties -- their Geometry & Combinatorics The conferences are supported by an NSF grant which allows us to provide a modest transportation allowance to those attendees who are not local. We also gratefully acknowledge support from Smith College and Wesleyan University. Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)math.smith.edu Karen Collins (Wesleyan University), (860) 685-2169, kcollins(at-sign)wesleyan.edu Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)math.smith.edu From sara(at-sign)math.mit.edu Wed Apr 22 15:58:57 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id PAA21257 for ; Wed, 22 Apr 1998 15:58:56 -0400 (EDT) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id PAA15718; Wed, 22 Apr 1998 15:58:56 -0400 (EDT) Date: Wed, 22 Apr 1998 15:58:56 -0400 (EDT) Message-Id: <199804221958.PAA15718(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Speaker: Andrzej Rucinski Title: On graphs with linear Ramsey numbers Abstract: For a fixed graph $H$, the \emph{Ramsey number} $r(H)$ is defined to be the least integer $N$ such that in any 2-coloring of the edges of the complete graph $K_{N}$, some monochromatic copy of $H$ is always formed. Let ${\mathcal H}(n,\Delta)$ denote the class of graphs $H$ having $n$ vertices and maximum degree $\Delta$. It was shown by Chvat\'al, R\"odl, Szemer\'edi and Trotter that for each $\Delta$ there exists $c(\Delta)$ such that $r(H) < c(\Delta)n$ for all $H \in {\mathcal H}(n,\Delta)$. That is, the Ramsey numbers grow \emph{linearly} with the size of $H$. However, their proof relied on the well-known regularity lemma of Szemeredi and only gave an upper bound for $c(\Delta)$ which grew like an exponential tower of $2's$ of height $\Delta$. In this talk we avoid the use of the regularity lemma altogether, and show that one can in fact take, for some fixed $c$, $c(\Delta) < 2^{c\Delta(\log \Delta)^{2}}$ in the general case, and even $c(\Delta) < 2^{c\Delta\log \Delta}$ if $H$ is bipartite. In particular, we improve an old upper bound on the Ramsey number of the $n$-cube due to Beck \cite{B}. We also show that for all $n$, there are $H' \in {\mathcal H}(n,\Delta)$ with $r(H') > 2^{c'\Delta}n$ for a fixed $c' > 0$, which is not so far from our upper bound. This is a joint work with R.L.Graham and V.R\"odl. Upcoming Events: April 24: Lou Shapiro Path pairs and Catalan probabilities April 29: Henrik Eriksson. Sorting bridge hands and DNA May 6:David Ingerman (NYU) Inverse boundary problems, arrangements of pseudo-lines and total positivity May 8: Christian Lenart (MIT) Combinatorial Aspects of the K-theory of flag varieties May 13: Jim Propp (MIT) Diabolo tilings of fortresses ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From sara(at-sign)math.mit.edu Fri Apr 24 13:08:49 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA15764 for ; Fri, 24 Apr 1998 13:08:49 -0400 (EDT) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id NAA15910; Fri, 24 Apr 1998 13:08:49 -0400 (EDT) Date: Fri, 24 Apr 1998 13:08:49 -0400 (EDT) Message-Id: <199804241708.NAA15910(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Speaker: Lou Shapiro Title: Path pairs and Catalan probabilities Abstract. Path pairs are pairs of paths starting at the origin, each consisting of unit east and north steps, with the upper path never going below the lower path, and with the two rapths having a common endpoint. They are in many ways the mother of all Catalan settings. Togrther with generating functions and k-motzkin paths we can derive probabalistic statements about many Catalan phenomena. For instance as n gets large the number of elements in the visible blocks of a noncrossing partition approaches 8. A variety of such results will be discussed including walking through an Eulerian polygonal maze. Upcoming Events: April 29: Henrik Eriksson. Sorting bridge hands and DNA May 6:David Ingerman (NYU) Inverse boundary problems, arrangements of pseudo-lines and total positivity May 8: Christian Lenart (MIT) ***Pretalk at 3:30 in 2-338*** Combinatorial Aspects of the K-theory of flag varieties May 13: Jim Propp (MIT) Diabolo tilings of fortresses ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From beers(at-sign)artemis.simmons.edu Fri Apr 24 15:55:02 1998 Return-Path: Received: from artemis.simmons.edu (artemis.simmons.edu [134.140.112.3]) by math.mit.edu (8.8.7/8.8.7) with SMTP id PAA21841 for ; Fri, 24 Apr 1998 15:55:02 -0400 (EDT) Received: from mathscs346.simmons.edu by artemis.simmons.edu (5.65v4.0/1.1.8.2/16Oct95-1036AM) id AA06577; Fri, 24 Apr 1998 15:55:13 -0400 Date: Fri, 24 Apr 1998 15:55:13 -0400 Message-Id: <3.0.16.19800104141233.35ef0d9a(at-sign)artemis.simmons.edu> X-Sender: beers(at-sign)artemis.simmons.edu X-Mailer: Windows Eudora Pro Version 3.0 (16) To: combinatorics(at-sign)math.mit.edu From: Donna Beers Subject: seminar today Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Speaker: Lou Shapiro Title: Path pairs and Catalan probabilities Abstract. Path pairs are pairs of paths starting at the origin, each consisting of unit east and north steps, with the upper path never going below the lower path, and with the two rapths having a common endpoint. They are in many ways the mother of all Catalan settings. Togrther with generating functions and k-motzkin paths we can derive probabalistic statements about many Catalan phenomena. For instance as n gets large the number of elements in the visible blocks of a noncrossing partition approaches 8. A variety of such results will be discussed including walking through an Eulerian polygonal maze. Upcoming Events: April 29: Henrik Eriksson. Sorting bridge hands and DNA May 6:David Ingerman (NYU) Inverse boundary problems, arrangements of pseudo-lines and total positivity May 8: Christian Lenart (MIT) ***Pretalk at 3:30 in 2-338*** Combinatorial Aspects of the K-theory of flag varieties May 13: Jim Propp (MIT) Diabolo tilings of fortresses ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From sara(at-sign)math.mit.edu Mon Apr 27 13:33:58 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA08798 for ; Mon, 27 Apr 1998 13:33:57 -0400 (EDT) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id NAA16091; Mon, 27 Apr 1998 13:33:57 -0400 (EDT) Date: Mon, 27 Apr 1998 13:33:57 -0400 (EDT) Message-Id: <199804271733.NAA16091(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: [daisymae(at-sign)math.mit.edu: Applied Mathematics Colloquium -- Monday, April 27, 1998] X-URL: http://www-math.mit.edu/~sara Just in case you are not on the Applied Math Colloquium list, here a talk you may be interested in. Date: Mon, 27 Apr 1998 11:15:03 -0400 (EDT) From: Shirley Entzminger-Merritt X-Sender: daisymae(at-sign)schauder To: amc(at-sign)math.mit.edu, gradstu(at-sign)math.mit.edu Subject: Applied Mathematics Colloquium -- Monday, April 27, 1998 T O D A Y APPLIED MATHEMATICS COLLOQUIUM TOPIC: COMBINATORICS AND TOPOLOGY OF GRAPH PROPERTIES SPEAKER: PROFESSOR ANDERS BJORNER Department of Mathematics Royal Institute of Technology Sweden DATE: MONDAY, APRIL 27, 1998 TIME: 4:15 P.M. LOCATION: Building 2, Room 105 Refreshments will be served at 3:45 p.m. in Building 2, Room 349 Applied Math Colloquium: http://www-math.mit.edu/amc/spring98 Math Department: http://www-math.mit.edu Massachusetts Institute of Technology Department of Mathematics Cambridge, MA 02139 From kcollins(at-sign)mail.wesleyan.edu Wed Apr 29 10:17:05 1998 Return-Path: Received: from mail.wesleyan.edu (dns.wesleyan.edu [129.133.12.10]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id KAA11319; Wed, 29 Apr 1998 10:17:01 -0400 (EDT) Received: from [129.133.30.205] (kcollins1.math.wesleyan.edu [129.133.30.205]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id KAA25489; Wed, 29 Apr 1998 10:06:58 -0400 (EDT) X-Sender: kcollins(at-sign)mail.wesleyan.edu Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 29 Apr 1998 10:17:52 -0400 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: third announcement Come to the Thirtieth one day conference on Combinatorics and Graph Theory Saturday, May 2, 1998 10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063 Schedule 10:00 Dan Rockmore (Dartmouth College) FFTs for $SL_2(F_q)$ - Theory and Applications 11:10 Randall McCutcheon (Wesleyan Univ.) Sets of recurrence in semigroups and Ramsey theory 12:10 Lunch 2:00 Joel Spencer (New York Univ. -- Courant Institute) Sixty Years of Ramsey R(3,k) 3:10 Lakshmibai (Northeastern Univ.) Flag varieties -- their Geometry & Combinatorics The conferences are supported by an NSF grant which allows us to provide a modest transportation allowance to those attendees who are not local. We also gratefully acknowledge support from Smith College and Wesleyan University. Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)math.smith.edu Karen Collins (Wesleyan University), (860) 685-2169, kcollins(at-sign)wesleyan.edu Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)math.smith.edu From sara(at-sign)math.mit.edu Wed Apr 29 13:17:23 1998 Return-Path: Received: from severi.mit.edu (SEVERI.MIT.EDU [18.87.0.68]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA17958; Wed, 29 Apr 1998 13:17:23 -0400 (EDT) Received: (from sara(at-sign)localhost) by severi.mit.edu (8.8.7/8.6.9) id NAA19398; Wed, 29 Apr 1998 13:17:22 -0400 (EDT) Date: Wed, 29 Apr 1998 13:17:22 -0400 (EDT) Message-Id: <199804291717.NAA19398(at-sign)severi.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Speaker: Henrik Eriksson. Title: Sorting bridge hands and DNA Abstract: Most bridge players start by sorting their hand, repeatedly moving blocks of cards. Open problem: Can this be done in at most seven moves? Solved problem: Sort the reversed permutation in seven moves! Block moves are fundamental mutations of DNA-molecules, so the block move distance can also be used for phylogenic estimates. The talk will review what is known about the problem and what we believe to be true. (Joint work with Kimmo Eriksson, KTH.) Upcoming Events: May 6:David Ingerman (NYU) Inverse boundary problems, arrangements of pseudo-lines and total positivity May 8: Christian Lenart (MIT) ***Pretalk at 3:30 in 2-338*** Combinatorial Aspects of the K-theory of flag varieties May 13: Jim Propp (MIT) Diabolo tilings of fortresses ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From iarrobin(at-sign)neu.edu Fri May 1 13:52:08 1998 Return-Path: Received: from NUHUB.DAC.NEU.EDU (nuhub.dac.neu.edu [129.10.1.6]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA19995 for ; Fri, 1 May 1998 13:52:08 -0400 (EDT) Received: from [155.33.184.186] by neu.edu (PMDF V5.1-8 #24746) with ESMTP id <01IWIS76KKXC8ZNADE(at-sign)neu.edu> for combinatorics(at-sign)math.mit.edu; Fri, 1 May 1998 13:52:15 EST Date: Fri, 01 May 1998 11:25:15 +0100 From: "a. Iarrobino" Subject: of possible interest X-Sender: iarrobin(at-sign)neu.edu (Unverified) To: combinatorics(at-sign)math.mit.edu Message-id: MIME-version: 1.0 Content-type: text/plain; charset="us-ascii" Combinatorics related talk at Northeastern GAS seminar: Monday May 4 at 1:30 PM Geometry-Algebra-Singularities Seminar: "Total positivity and double Bruhat cells" Andrei Zelevinsky (Northeastern) Northeastern University, 509 Lake Hall Abstract: An invertible matrix is totally nonnegative if all its minors (including all matrix entries) are nonnegative real numbers. We discuss the generalization (due to G. Lusztig) of this classical notion to any semisimple group. The natural geometric framework for this study is provided by the decomposition of the group into a disjoint union of double Bruhat cells (intersections of cells in two Bruhat decompositions with respect to opposite Borel subgroups. This is joint work with S. Fomin From propp(at-sign)math.mit.edu Mon May 4 12:30:23 1998 Return-Path: Received: from pfaff.mit.edu (PFAFF.MIT.EDU [18.87.0.183]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id MAA16637 for ; Mon, 4 May 1998 12:30:23 -0400 (EDT) Received: (from propp(at-sign)localhost) by pfaff.mit.edu (8.8.7/8.6.9) id MAA00265 for combinatorics(at-sign)math.mit.edu; Mon, 4 May 1998 12:30:12 -0400 (EDT) Date: Mon, 4 May 1998 12:30:12 -0400 (EDT) From: Jim Propp Message-Id: <199805041630.MAA00265(at-sign)pfaff.mit.edu> To: combinatorics(at-sign)math.mit.edu Subject: Discrete Dinner The Boston area D I S C R E T E D I N N E R will be held at P H O P A S T E U R (a Vietnamese restaurant at 35 Dunster Street in Harvard Square) at 6 : 0 0 P M on W E D N E S D A Y , M A Y 1 3 R.S.V.P. to Jim Propp (propp(at-sign)math.mit.edu) by Friday, so that we can make a reservation for a table of suitable size. (If the number of people in your party is a random variable, please give its expected value.) See you then, Jim Propp From sara(at-sign)math.mit.edu Wed May 6 15:07:12 1998 Return-Path: Received: from schubert.mit.edu (SCHUBERT.MIT.EDU [18.87.0.16]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id PAA28282 for ; Wed, 6 May 1998 15:07:11 -0400 (EDT) Received: (from sara(at-sign)localhost) by schubert.mit.edu (8.8.7/8.8.7) id PAA12181; Wed, 6 May 1998 15:07:06 -0400 (EDT) Date: Wed, 6 May 1998 15:07:06 -0400 (EDT) Message-Id: <199805061907.PAA12181(at-sign)schubert.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 Speaker: David Ingerman (NYU, ingerman(at-sign)boheme.cims.nyu.edu) Title: Inverse boundary problems, arrangements of pseudo-lines and total positivity Abstract: Abstract: Let $\Gamma$ be a planar graph embedded in a closed disk. Each positive weight function $\gamma$ on the edges of $\Gamma$ induces a response map of the graph (the linear map from Dirichlet to Neumann boundary data of $\gamma$-harmonic functions.) We will consider inverse boundary problems of obtaining information about the weighted graph from the response map. Our motivation for the study of these problems comes from their continuous PDE analogs (and originally from oil production industry). Surprisingly the answers turn out to be combinatorial in nature. For example: Every weighted graph is determined by its response matrix up to simple Reidemeister type transformations. The quotient of the weighted graphs by the transformations is essentially the set of circular permutations. There is a natural one-to-one correspondence between the graphs for which the inverse problem has unique solution and the arrangements of pseudo-lines on the plane. The response matrices are completely characterized by their total positivity property (sign conditions on determinants of minors.) Upcoming Events: May 8: Christian Lenart (MIT, lenart(at-sign)math.mit.edu) Combinatorial Aspects of the K-theory of flag varieties ***Pretalk at 3:30 in 2-338*** May 13: Jim Propp (MIT, propp(at-sign)math.mit.edu) Diabolo tilings of fortresses ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From sara(at-sign)math.mit.edu Fri May 8 13:42:22 1998 Return-Path: Received: from schubert.mit.edu (SCHUBERT.MIT.EDU [18.87.0.16]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id NAA00181 for ; Fri, 8 May 1998 13:42:22 -0400 (EDT) Received: (from sara(at-sign)localhost) by schubert.mit.edu (8.8.7/8.8.7) id NAA25035; Fri, 8 May 1998 13:42:21 -0400 (EDT) Date: Fri, 8 May 1998 13:42:21 -0400 (EDT) Message-Id: <199805081742.NAA25035(at-sign)schubert.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 in 2-338********** refreshments served at 3:45 ***Pretalk at 3:30 in 2-338*** Speaker: Christian Lenart (MIT, lenart(at-sign)math.mit.edu) Title: Combinatorial Aspects of the K-theory of flag varieties Abstract: In this talk we present some results concerning Grothendieck polynomials, which are representatives for the classes corresponding to Schubert varieties in the K-theory of the flag variety. Grothendieck polynomials are nonhomogeneous polynomials whose lowest homogeneous component is the Schubert polynomial indexed by the same permutation; hence they offer more information about the geometry of the flag variety than Schubert polynomials. The first part of the talk is devoted to Grothendieck polynomials corresponding to Grassmannian permutations; we discuss their expansion in the basis of Schur polynomials and a Pieri rule. In the second part of the talk, we sketch the proof of a conjecture of A. Lascoux concerning the expansion of an arbitrary Grothendieck polynomial in the basis of Schubert polynomials, and present a combinatorial interpretation for the coefficients of this expansion. The proof is based on certain noncommutative analogs of Schubert polynomials, for which we prove a Pieri rule and a Cauchy identity, thus extending the work of S. Fomin and C. Greene on noncommutative Schur functions. Upcoming Events: May 13: Jim Propp (MIT, propp(at-sign)math.mit.edu) Diabolo tilings of fortresses Happy Summer Vacation, we will continue the combinatorics seminar on Wednesday, September 9th. ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From sara(at-sign)math.mit.edu Wed May 13 12:50:02 1998 Return-Path: Received: from schubert.mit.edu (SCHUBERT.MIT.EDU [18.87.0.16]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id MAA28589 for ; Wed, 13 May 1998 12:50:01 -0400 (EDT) Received: (from sara(at-sign)localhost) by schubert.mit.edu (8.8.7/8.8.7) id MAA20748; Wed, 13 May 1998 12:50:00 -0400 (EDT) Date: Wed, 13 May 1998 12:50:00 -0400 (EDT) Message-Id: <199805131650.MAA20748(at-sign)schubert.mit.edu> From: Sara Billey To: combinatorics(at-sign)math.mit.edu Subject: seminar today X-URL: http://www-math.mit.edu/~sara *********Combinatorics Seminar Today at 4:15 ********** refreshments served at 3:45 ***NOTE: Seminar will be held in 2-131*** Speaker: Jim Propp (MIT, propp(at-sign)math.mit.edu) Title: Diabolo tilings of fortresses Abstract: Diabolo tilings of fortresses are combinatorial objects that share some features with domino tilings of Aztec diamonds, rhombus tilings of hexagons, and alternating sign matrices, but that also exhibit novel phenomena that are not yet entirely understood. After defining these tilings, I will present Mihai Ciucu's simple proof of Bo-Yin Yang's formula for the number of diabolo tilings of the fortress of order n, using the graph-theoretic method of "urban renewal". Then I will show how extensions of urban renewal allow one to efficiently sample uniformly from the set of such tilings, and to encode statistical properties of random tilings via generating functions. These techniques suggest, and should eventually allow us to prove, that in addition to (homogeneous) "frozen regions" and a (non-homogeneous) "liquid region", diabolo tilings of fortresses also exhibit "gaseous" behavior in the center of the tiling, as seen in "http://www-math.mit.edu/~propp/hidden/fort200.ps.gz" a random diabolo tiling of a fortress of order 200 and "http://www-math.mit.edu/~propp/hidden/fort300.5.ps.gz" a color-plot of the local statistics for the ensemble of random diabolo tilings of a fortress of order 300. Other displays used in this talk can be found at "http://www-math.mit.edu/~propp/diabolo.ps.gz" This is joint work with Mihai Ciucu, Henry Cohn, Rick Kenyon, David Wilson, and the undergraduate members of the Tilings Research Group. The talk presupposes no knowledge of tiling theory or statistical mechanics. Note that the talk will take place in room 2-131 (not the usual meeting place for the seminar). Upcoming Events: Discrete Dinner tonight. Happy Summer Vacation, we will continue the combinatorics seminar on Wednesday, September 9th. ******************************************************* * Sara Billey * * Applied Mathematics Instructor * * Massachusetts Institute of Technology * * Web page: http://www-math.mit.edu/~sara/ * * Room: 2-334 Phone: (617)-253-7775 * ******************************************************* From daisymae(at-sign)math.mit.edu Tue Jun 9 13:42:30 1998 Return-Path: Received: from schauder (SCHAUDER.MIT.EDU [18.87.0.69]) by math.mit.edu (8.8.7/8.8.7) with SMTP id NAA26606 for ; Tue, 9 Jun 1998 13:42:30 -0400 (EDT) Date: Tue, 9 Jun 1998 13:42:29 -0400 (EDT) From: Shirley Entzminger-Merritt X-Sender: daisymae(at-sign)schauder Reply-To: Shirley Entzminger-Merritt To: combinatorics(at-sign)math.mit.edu Subject: Special Lecture -- Friday, June 12, 1998 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII COMBINATORICS SEMINAR SPEAKER: Ezra Miller University of California, Berkeley TITLE: Alexander duality for arbitrary monomial ideals DATE: Friday, June 12, 1998 TIME: 2:00 p.m. LOCATION: MIT Building 2, Room 338 Abstract: Given a simplicial complex $X$ on a finite vertex set, the {\it Alexander dual} $X^\vee$ of $X$ is another simplicial complex (on the same vertex set) whose homology is isomorphic to the cohomology of $X$. This construction carries over to the corresponding face ideals $I$ and $I^\vee$, which are squarefree monomial ideals in a polynomial ring. The talk will generalize the construction of $I^\vee$ from $I$ to the nonsquarefree case. Several equivalent constructions of the {\it Alexander dual ideal} will be outlined, relating duality in various lattices to duality in regular cell complexes and the switching of minimal generators for irreducible components of an ideal. From daisymae(at-sign)math.mit.edu Wed Jun 10 12:58:01 1998 Return-Path: Received: from schauder (SCHAUDER.MIT.EDU [18.87.0.69]) by math.mit.edu (8.8.7/8.8.7) with SMTP id MAA15855 for ; Wed, 10 Jun 1998 12:58:01 -0400 (EDT) Date: Wed, 10 Jun 1998 12:58:01 -0400 (EDT) From: Shirley Entzminger-Merritt X-Sender: daisymae(at-sign)schauder To: combinatorics(at-sign)math.mit.edu Subject: Special Lecture -- Friday, June 12, 1998 -- CANCELLED In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII NOTE: THIS SEMINAR HAS BEEN CANCELLED. COMBINATORICS SEMINAR SPEAKER: Ezra Miller University of California, Berkeley TITLE: Alexander duality for arbitrary monomial ideals DATE: Friday, June 12, 1998 -- CANCELLED TIME: 2:00 p.m. LOCATION: MIT Building 2, Room 338 From kcollins(at-sign)mail.wesleyan.edu Tue Jul 7 16:31:47 1998 Return-Path: Received: from mail.wesleyan.edu (dns.wesleyan.edu [129.133.12.10]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id QAA06116; Tue, 7 Jul 1998 16:31:47 -0400 (EDT) Received: from [129.133.30.205] (kcollins1.math.wesleyan.edu [129.133.30.205]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id QAA11836; Tue, 7 Jul 1998 16:31:13 -0400 (EDT) X-Sender: kcollins(at-sign)mail.wesleyan.edu Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Tue, 7 Jul 1998 16:37:17 -0400 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: Montreal announcement Therese Biedl would like everyone to know about the following conference. Final call for POSTERS Deadline: July 15, 1998 Graph Drawing '98 McGill University, Montreal, Canada, August 13-15, 1998. URL: http://gd98.cs.mcgill.ca email: info(at-sign)gd98.cs.mcgill.ca The paper submission deadline for GD '98 has now passed, and the list of accepted papers is available at the above Web site. However, poster submissions for GD '98 are still welcome. Graph Drawing '98 (GD '98) will be held at McGill University, Montreal, Canada, August 13 - 15, 1998. The symposium is a forum for researchers, practitioners, developers and users working on all aspects of graph drawing. GD '98 follows the 10th Canadian Conference on Computational Geometry, August 10-12, 1998, held at McGill University. See http://cgm.cs.mcgill.ca/cccg98. Call for Posters: Submissions of posters in graph drawing and related areas are solicited. The purpose of posters is to provide a forum for the communication of results to the graph drawing community. These posters may contain results that have appeared or will appear elsewhere. They may also pose open problems of interest to the graph drawing community. To increase the interaction between graph drawing and other areas, posters that present topics related to graph drawing in fields such as cartography, chemistry, computational biology, geographic information systems, graphics, perception and vision, scientific visualization, and software engineering are particularly encouraged. To allow for questions and discussion, at least one author of each poster is expected to be present at the conference. Proceedings: The proceedings of GD '98 will be published in the Springer-Verlag Lecture Notes in Computer Science series. These proceedings will contain a Poster Gallery Report, with a 1-2 page abstract per poster. The camera-ready version of this abstract is due at the conference. To prepare your hard copy, please follow the directions for authors in the LNCS series of Springer-Verlag. These directions are available at the LNCS homepage. Submission: To submit a poster, send a 1-2 page abstract by email (in postscript, LaTeX, or plain text) or as hard copy to the Poster Chair Therese Biedl School of Computer Science McGill University 3480 University Street #318 Montreal, Quebec H3A 2A7, Canada email: therese(at-sign)cs.mcgill.ca Important dates: Submissions: July 15, 1998. Final version: August 15, 1998. From kcollins(at-sign)mail.wesleyan.edu Sun Aug 16 11:17:29 1998 Return-Path: Received: from mail.wesleyan.edu (mail.wesleyan.edu [129.133.1.51]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id LAA18338; Sun, 16 Aug 1998 11:17:29 -0400 (EDT) Received: from [129.133.30.205] (kcollins1.math.wesleyan.edu [129.133.30.205]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id LAA19342; Sun, 16 Aug 1998 11:15:29 -0400 (EDT) X-Sender: kcollins(at-sign)mail.wesleyan.edu Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Sun, 16 Aug 1998 11:23:39 -0400 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: First CoNE meeting this year Come to the Thirty-first one day conference on Combinatorics and Graph Theory Saturday, September 12, 1998 10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063 Schedule 10:00 Elizabeth McMahon (Lafayette College) TBA 11:10 Lowell Beineke (Indiana Univ. -- Purdue Univ. Fort Wayne) Creating and Destroying Cycles in Graphs and Digraphs** 12:10 Lunch 2:00 Louis Billera (Cornell University) TBA 3:10 TBA **See abstract below** The conferences are supported by an NSF grant which allows us to provide a modest transportation allowance to those attendees who are not local. We also gratefully acknowledge support from Smith College and Wesleyan University. Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)math.smith.edu Karen Collins (Wesleyan University), (860) 685-2169, kcollins(at-sign)wesleyan.edu Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)math.smith.edu ** ** Creating and Destroying Cycles in Graphs and Digraphs Lowell Beineke,Indiana Univ. -- Purdue Univ. Fort Wayne We will discuss results on two topics involving cycles in graphs. The first type of problem is to determine the maximum number of cyclic triples among all oriented graphs with some prescribed property. Our focus will be on oriented graphs with a fixed number of arcs (the result for a fixed number of vertices is well known). Except when the number of vertices is small in comparison with the number of arcs, we have a complete solution. We will also look at planar graphs in this context. At the opposite extreme, there is the problem of eliminating all cycles efficiently, such as through the removal of vertices or arcs. We will consider this problem for various families of graphs and digraphs. From kcollins(at-sign)mail.wesleyan.edu Sun Aug 23 11:30:28 1998 Return-Path: Received: from mail.wesleyan.edu (mail.wesleyan.edu [129.133.1.51]) by math.mit.edu (8.8.7/8.8.7) with ESMTP id LAA16758; Sun, 23 Aug 1998 11:30:26 -0400 (EDT) Received: from [129.133.30.205] (kcollins1.math.wesleyan.edu [129.133.30.205]) by mail.wesleyan.edu (8.8.6/8.7.3) with SMTP id LAA24147; Sun, 23 Aug 1998 11:27:47 -0400 (EDT) X-Sender: kcollins(at-sign)mail.wesleyan.edu Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Sun, 23 Aug 1998 11:35:47 -0400 To: kcollins(at-sign)mail.wesleyan.edu From: kcollins(at-sign)wesleyan.edu (Karen L. Collins) Subject: Sept. 12th, second annoucement Come to the Thirty-first one day conference on Combinatorics and Graph Theory Saturday, September 12, 1998 10 a.m. to 4:30 p.m. at Smith College Northampton MA 01063 Schedule 10:00 Elizabeth McMahon (Lafayette College) TBA 11:10 Lowell Beineke (Indiana Univ. -- Purdue Univ. Fort Wayne) Creating and Destroying Cycles in Graphs and Digraphs 12:10 Lunch 2:00 Louis Billera (Cornell University) Flag Enumeration in Convex Polytopes 3:10 Kenneth Bogart (Dartmouth College) Trapezoid and Parallelogram Graphs and Orders The conferences are supported by an NSF grant which allows us to provide a modest transportation allowance to those attendees who are not local. We also gratefully acknowledge support from Smith College and Wesleyan University. Our Web page site has directions to Smith College, abstracts of speakers, dates of future conferences, and other information. The address is: http://math.smith.edu/~rhaas/coneweb.html Michael Albertson (Smith College), (413) 585-3865, albertson(at-sign)math.smith.edu Karen Collins (Wesleyan University), (860) 685-2169, kcollins(at-sign)wesleyan.edu Ruth Haas (Smith College), (413) 585-3872, rhaas(at-sign)math.smith.edu