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
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.
Resources in the Division of Student Life.
Support in the Office of Graduate Education.
A partial list of appropriate papers can be found
Niven Achenjang firstname.lastname@example.org
Jordan Benson email@example.com
Deeparaj Bhat firstname.lastname@example.org
Cameron Krulewski email@example.com
David Jongwon Lee firstname.lastname@example.org
Jae Lee email@example.com
Junyao Peng firstname.lastname@example.org
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 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: 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
Department of Mathematics 2-478
Massachusetts Institute of Technology
Cambridge, MA 02139
Zoom office: https://mit.zoom.us/j/6691725321