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 Dan Kan. 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.
  • Graduate Student Resources in the Division of Student Life.
  • Graduate Student Support in the Office of Graduate Education.

    A partial list of appropriate papers can be found here.

    Participants

    Zihong Chen zihongch@mit.edu
    Preston Cranford prestonc@mit.eu
    Haoshuo Fu fuhs@mit.edu
    Swapnil Garg swapnilg@mit.edu
    Carina Letong Hong clhong@mit.edu
    Serina Hu serinahu@mit.edu
    Gabrielle Yangqing Li gabbyyli@mit.edu
    Weixiao Lu weixiaol@mit.edu
    Mikayel Mkrtchyan mikayelm@mit.edu
    Dylan Pentland dylanp@mit.edu
    Natalie Stewart nataliestewart@math.harvard.edu
    Eunice Sukarto esukarto@math.harvard.edu
    Tristan Yang tristanyang@college.harvard.edu

    Schedule
    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 d'Eilenberg-MacLane
    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 espaces homogenes
    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 differentiables
    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 analogue
    Fri 15 Oct: Carina Hong: Dyer and Lashof, A topological proof of the Bott periodicity theorem
    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 ring
    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 homology
    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 cobordism
    Fri 3 Dec: Tristan Yang: Devinatz, Hopkins, Smith: Nilpotence and stable homotopy theory I
    Mon 6 Dec: No meeting
    Wed 8 Dec:


    Some resources

    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 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: 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 https://arxiv.org instead.

    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.

    Haynes Miller
    Department of Mathematics 2-478
    Massachusetts Institute of Technology
    Cambridge, MA 02139
    Email: hrm@math.mit.edu
    Zoom office: https://mit.zoom.us/j/6691725321
    Website: math.mit.edu/~hrm

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