18.919: Seminar on Algebraic Topology


Monday, Wednesday, Friday, 11:00, 2-255



This is a literature seminar with a focus on classic papers in Algebraic Topology. Each student will give one or two talks on each of three papers, chosen in consultation with me, and will also read all the papers talked on in a more cursory way, and write brief reactions to them. This course will prove useful not only to students intending to pursue algebraic topology, but also to those interested in symplectic geometry, algebraic geometry, representation theory, and combinatorics.
A good description of the underlying intent of the seminar has been given by Phil Hirschhorn:
"I think the most important function of the Kan Seminar was to push students through the transition from someone who takes courses to someone who thinks more actively about mathematics. You had to make sense of what had been done in those seminal papers, and try to explain that to other students (and to Dan, who was a very critical audience). Speaking in that seminar was a somewhat terrifying experience for most of us, which is why Dan never allowed visitors; if the speaker felt stressed and embarrassed, at least it was only in front of others going through the same thing."

A partial list of appropriate papers can be found here.
I will want to interview prospective participants early in the semester or before it starts.

Some resources

Most of the papers we will read are available online, through the MIT library's VERA database. You need an MIT certificate to use it. Many journals are also available directly through MathSciNet. Here are some other sources. From VERA you can get to JSTOR, a huge archive of journal arcticles from all disciplines, or to individual journals.

For mainly German documents visit Goettinger Digitalisierungs-Zentrum and follow links to Mathematical Literature.

For much earlier work, try la bibliotheque Gallica-Math. Maybe the most useful link from there is to NUMDAM, an archive of seminars and other mathematical documents. I especially commend to you the Seminaire Henri Cartan:
Year 1950-51: Cohomologie des groupes, suite spectrale, faisceaux
Years 1953-55: Algebre d'Eilenberg-Maclane et homotopie
Year 1958-59: Invariant de Hopf et operations cohomologiques secondaires
Year 1959-60: Periodicite des groupes d'homotopie stable des groupes classiques, d'apres Bott

For more recent work, the standard preprint server is the Front for the Mathematics ArXiv.


Participants

Alexey Balitskiy, balitski@mit.edu
Yongyi Chen, yongyic@mit.edu
Sanath Devalapurkar, sanathd@mit.edu
Panagiotis Dimakis, pdimakis@mit.edu
Jesse Freeman, freemanj@mit.edu
Araminta Gwynne, araminta@mit.edu
Dmitry Kubrak, dmkubrak@mit.edu
Andrei Ionov, aionov@mit.edu
Zhulin Li, zhulin@mit.edu
Morgan Opie, opie.morgan@gmail.com
Juan Ortiz, jor@mit.edu
Oron Propp, opropp@mit.edu
Luke Sciarappa, lukesci@mit.edu
Sahana Vasudevan, sahanav@mit.edu
Miguel Young, mcyoung@mit.edu

Schedule

Wed 6 Sep : Organization.
Fri 8 Sep : --
Mon 11 Sep : Zhulin Li : Serre, Cohomology des complexes d'Eilenberg Mac Lane. For background on Steenrod operations see these notes, pp 38--58, Hatcher's AT, or Spanier (who does things at the chain level).
Wed 13 Sep : --
Fri 15 Sep : Jesse Freeman : Thom, Quelques proprietes globales des varietes differentiables. Very poor English translation is here.
Mon 18 Sep : --
Wed 20 Sep : Juan Ortiz : same paper, continued.
Fri 22 Sep : Panagiotis Dimakis : Milnor and Stasheff, Characteristic classes (The signature theorem).
Mon 25 Sep : Sanath Devalapurkar : Milnor, On manifolds homeomorphic to the 7-sphere.
Wed 27 Sep : Miguel Young : Kan, A combinatorial definition of homotopy groups.
Mon 2 Oct : Dmitry Kubrak : Brown, Cohomology theories.
Wed 4 Oct : Oron Propp : Atiyah, K-theory.
Fri 6 Oct : Luke Sciarappa : Adams and Atiyah, K-theory and the Hopf invariant.
Wed 11 Oct : Andrei Ionov : Adams, Vector fields on spheres.
Fri 13 Oct : Sahana Vasudevan : Smale, Classification of immersions of spheres into Euclidean spaces.
Mon 16 Oct : --
Wed 18 Oct : Alexey Balitskiy : Kervaire and Milnor, Groups of homotopy spheres.
Fri 20 Oct : Yongyi Chen : Douady, Periodicite du group unitaire.
Mon 23 Oct : Morgan Opie : Adams, J(X) IV.
Wed 25 Oct : Panagiotis Dimakis : Adams, Stable homotopy and generalized homology.
Fri 27 Oct : Araminta Gwynne : Becker and Gottlieb, The transfer map and fiber bundles.
Mon 30 Oct : --
Wed 1 Nov : Zhulin Li : Quillen, The spectrum of an equivariant cohomology ring.
Fri 3 Nov : Jesse Freeman : May, Geometry of Iterated Loop Spaces.
Mon 6 Nov : Juan Ortiz : Atiyah and Segal, Equivariant K-theory and completion.
Wed 8 Nov : Dmitri Kubrak : Segal, Categories and cohomology theories.
Mon 13 Nov : --
Wed 15 Nov : Sahana Vasudevan : Quillen, Homotopical Algebra.
Fri 17 Nov : Luke Sciarappa : Bousfield, Localization of spaces with respect to homology.
Mon 20 Nov : Andrei Ionov : Griffiths and Morgan, Rational homology theory and differential forms.
Wed 22 Nov : --
Mon 27 Nov : Oron Propp : Quillen, Higher algebraic K-theory, I.
Wed 29 Nov : Yongyi Chen : Milnor, On the cobordism ring \Omega_* and a complex analogue.
Fri 1 Dec : Alexey Balitskiy : Karoubi, Cobordisme et groupes formels (d'apres D. Quillen et T. tom Dieck)
Mon 4 Dec : --
Wed 6 Dec : Morgan Opie : Quillen, On the cohomology and K-theory of the general linear groups over a finite field.
Fri 8 Dec : Araminta Gwynne : Waldhausen, Algebraic K-theory of spaces.
Mon 11 Dec : Miguel Young : Carlsson, Equivariant stable homotopy theory and Segal's Burnside ring conjecture.
Wed 13 Dec : Sanath Devalapurkar : Ravenel, Localization with respect to certain periodic homology theories.

This paper by Steve Mitchell puts a lot of the ideas we've seen in this seminar into a common context.

The banquet



Haynes Miller
2-478, 3-7569
hrm@math.mit.edu