18.919: Seminar on Algebraic Topology
Monday, Wednesday, Friday, 11:00, in 2-255
This is a literature seminar with a focus on classic papers
in Algebraic Topology. Each student will give a talk 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.
This seminar was founded by the late
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."
The seminar is also described on OpenCourseWare.
There are two important components of this class beyond the seminar lectures.
(1) I expect a "reading response" to each paper that you are not reporting on.
It is due before the lecture on the paper. The idea is for you to
capture your thoughts about the paper: what struck you most strongly, how
does it fit with other mathematics you know, what other questions does it raise
for you? I will try to respond to each of these comments.
(2) I hope each lecture is preceded by a "practice talk." This is for you
to schedule, and attend as you like. History shows this practice to be
extremely valuable. I leave it to you to organize these.
If you are an undergraduate interested in pursuing this course, I would like
to speak with you in person before the term begins, to discuss what's involved
and whether this is a good choice for you.
Here are some resources available to you as MIT graduate students.
Resources in the Division of Student Life.
Support in the Office of Graduate Education.
A partial list of appropriate papers can be found
Zihong Chen firstname.lastname@example.org
Preston Cranford email@example.com
Haoshuo Fu firstname.lastname@example.org
Swapnil Garg email@example.com
Carina Letong Hong firstname.lastname@example.org
Serina Hu email@example.com
Gabrielle Yangqing Li
Weixiao Lu firstname.lastname@example.org
Mikayel Mkrtchyan email@example.com
Dylan Pentland firstname.lastname@example.org
Wed 8 Sep: Organizational meeting
Fri 10 Sep: No meeting
Mon 13 Sep: No meeting
Wed 15 Sep: Gabrielle Li: Serre, Cohomologie modulo 2 des complexes
Fri 17 Sep: Weixiao Lu: ditto
Mon 20 Sep: Zihong Chen: Moore,
Semi-simplicial complexes and Postnikov systems
Wed 22 Sep: Dylan Pentland: Borel, La cohomologie modulo 2 de certains
Fri 24 Sep: Mikayel Mkrtchyan: Milnor, The Steenrod algebra and its dual
Mon 27 Sep: No meeting
Wed 29 Sep: Preston Cranford: Thom, Quelques proprietes globales des varieties
Fri 1 Oct: Swapnil Garg, ditto
Mon 4 Oct: Haoshuo Fu: Hirzebruch, Topological methods in algebraic geometry
(the signature theorem)
Wed 6 Oct: Natalie Stewart: Milnor, On manifolds homeomorphic to the 7-sphere
Fri 8 Oct: Serina Hu: Brown, Cohomology theories
Wed 13 Oct: Tristan Yang: Milnor, On the cobordism ring and a complex
Fri 15 Oct: Carina Hong: Dyer and Lashof, A topological proof of the Bott
Mon 18 Oct: No meeting
Wed 20 Oct: Eunice Sukarto: Atiyah, K-theory
Fri 22 Oct: Gabrielle Li: Adams and Atiyah, K-theory and the Hopf invariant
Mon 25 Oct: No meeting
Wed 27 Oct:Weixiao Lu: Quillen, The spectrum of an equivariant cohomology
Fri 29 Oct: Dylan Pentland: Quillen, The cohomology and K-theory of the
general linear groups over a finite field
Mon 1 Nov: No meeting
Wed 3 Nov: Zihong Chen: Quillen, Higher Algebaic K-theory I
Fri 5 Nov: Mikayel Mkrtchyan: Griffiths and Morgan, Rational homotopy theory
and differential forms
Mon 8 Nov: No meeting
Wed 10 Nov: Preston Cranford: Quillen, Homotopical Algebra
Fri 12 Nov: Swapnil Garg: Hill, Hopkins, Ravenel, On the nonexistence of
elements of Kervaire invariant one (Orthogonal spectra)
Mon 15 Nov: No meeting
Wed 17 Nov: Haoshuo Fu: Bousfield, Localization of spectra with respect to
Fri 19 Nov: Natalie Stewart: May, The Geometry of Iterated Loop Spaces
Mon 22 Nov: Carina Hong: Adams, Vector fields on spheres
Wed 24 Nov: No meeting
Mon 29 Nov: No meeting
Wed 1 Dec: Serina Hu: Adams, Quillen's work on formal groups and complex
Fri 3 Dec: Tristan Yang: Devinatz, Hopkins, Smith: Nilpotence and stable
homotopy theory I
Mon 6 Dec: No meeting
Wed 8 Dec:
Some of the material at the start of the course is related to the end
of 18.906, at least as I gave it last term.
Here are lecture notes.
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
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
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: Algebres d'Eilenberg-Mac Lane 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,
though it seems to be out of commission right now, so use
Graeme Segal's description of Michael Atiyah's contributions
to topology, here, has just been
published in the Bulletin of the AMS. It contains beautiful
accounts of many of the
topics discussed in the first part of this seminar, emphasizing Atiyah's
deep contributions to them.
Department of Mathematics 2-478
Massachusetts Institute of Technology
Cambridge, MA 02139
Zoom office: https://mit.zoom.us/j/6691725321