18.919: Seminar on Algebraic Topology

Monday, Wednesday, Friday, 11:00, on Zoom

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.

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.

The Fall 2020 seminar will meet synchronously on Zoom, at https://mit.zoom.us/j/94193272880, meeting id 941 9327 2880. You will need to have a tablet and stylus. I expect most lectures will be supported by freehand slides, either premade or written on the fly. Attendence at these lectures is expected, and it would wonderful if all of us had the video on. This contributes to a sense of community. I do not intend to record these talks.

There are two other important components of this class.
(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.

This will be a difficult fall for everybody, posing many unfamiliar challenges. I hope we can all be patient and learn from each other. It's important that you let me know of problems that may present themselves, that might suggest changing schedules in one way or another for example. In addition to that, 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.


    Niven Achenjang nivent@mit.edu
    Jordan Benson jbenson@mit.edu
    Deeparaj Bhat deeparaj@mit.edu
    Cameron Krulewski camkru@mit.edu
    David Jongwon Lee jongwonl@mit.edu
    Jae Lee jaehelee@mit.edu
    Jiakai Li jiakaili@math.harvard.edu
    Junyao Peng junyaop@mit.edu
    Elia Portnoy portnoy.elia@gmail.com

    Tentative Schedule

    Wed 2 Sep: Organizational meeting
    Fri 4 Sep: no meeting
    Wed 9 Sep: Cameron: Serre, "Cohomologie modulo 2 ..."
    Fri 11 Sep: Jiakai: Borel, "La cohomologie modulo 2 ..."
    Mon 14 Sep: no meeting
    Wed 16 Sep: Deeparaj: Dyer and Lashof, "A topological proof ..."
    Fri 18 Sep: Jae: Thom, "Quelques proprietes ..."
    Mon 21 Sep: Jordan: Atiyah, "Bordism and cobordism"
    Wed 23 Sep: no meeting
    Fri 25 Sep: Elia: Hirzebruch, "Topological Methods ..."
    Mon 28 Sep: no meeting
    Wed 30 Sep: Junyao: Milnor, "On manifolds ..."
    Fri 2 Oct: Niven: Brown, "Cohomology theories"
    Mon 5 Oct: no meeting
    Wed 7 Oct: Jiakai, Atiyah, "K-theory"
    Fri 9 Oct: David: Adams, "Vector fields on spheres"
    Wed 14 Oct: David: ditto
    Fri 16 Oct: no meeting
    Mon 19 Oct: Deeparaj: May, "The geometry of iterated loop spaces"
    Wed 21 Oct: no meeting
    Fri 23 Oct: Jae: Quillen, "Spectrum of an equivariant ..."
    Mon 26 Oct: Cameron: Quillen, "Cohomology and K-theory ..."
    Wed 28 Oct: no meeting
    Fri 30 Oct: Jordan: Bousfield, "Localization of spaces ..."
    Mon 2 Nov: Elia: Griffiths and Morgan, "Rational homotopy theory ..."
    Wed 4 Nov: no meeting
    Fri 6 Nov: Junyao: Milnor, "On the cobordism ring ..."
    Mon 9 Nov: Niven: Adams, "Quillen's work..."
    Fri 13 Nov: David: Quillen, "Higher algebraic K-theory"
    Mon 16 Nov: Jiakai: Quillen, "Homotopical Algebra"
    Wed 18 Nov: no meeting
    Fri 20 Nov: Jae: Atiyah and Segal, "Equivariant K-theory ..."
    Mon 30 Nov: Jordan: Bousfield, "Localization of spectra ..."
    Wed 2 Dec: no meeting
    Fri 4 Dec: Junyao: Bousfield and Kan, "Homotopy limits ..."
    Mon 7 Dec: Niven: Morava, "Forms of $K$-theory"
    Wed 9 Dec: David: Morel and Voevodsky, "A^1-homotopy theory of schemes"

    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: 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, though it seems to be out of commission right now, so use https://arxiv.org instead.

    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