Proseminar on algebraic topology Fall 2005

Instructors: Daniel Biss, Jesper Grodal, Peter May, Nathalie Wahl, ...
Website: www.math.uchicago.edu/~jg/prosem.html
Time and place: We meet Tuesday and Thursday 13:30-15:00 in Eckhart Room 203
Audience: Graduate students in their second year or higher, with an interest in algebraic topology.
Textbooks: We'll use various books and research articles as detailed in the course plan below.

Various practical information and course plan:

Here is a list of topics which which we started on last year. This year we'll try to take some of these topics in more depth, in modules of 2 weeks (4 talks) per topic. If you are interested in giving a talk about one of them, drop us a line.

You should sign up for the mailing list "topology" by going to http://zaphod.uchicago.edu:8080/mailman/listinfo/topology

UPCOMING TALKS:

First module: Topological K-theory with a focus on the real case.
Thur Sept 29: Recollections on K-theory, Adams operations, and Hopf Invariant one (Peter May)
Tues Oct 4: Real K-theory and the image of J (Peter May)
Thur Oct 6: Peter May cont.
Tues Oct 11: Clifford algebras and KO-theory following Atiyah-Bott-Shapiro (Justin Noel)
Thur Oct 13: Real Bott periodicity following paper of Mark Behrens I (Niles Johnson)
Tues Oct 18: Real Bott periodicity following paper of Mark Behrens II (Niles Johnson)
Thur Oct 20: Real Bott periodicity following paper of Mark Behrens III (Niles Johnson)
Tues Oct 25: Vector fields on spheres (Megan Guichard)

Second module: Algebraic K-theory.
Thur Oct 27: Algebraic K-theory via the Waldhausen construction (Steffen Sagave)
Tues Nov 1: Definition of K-theory via the + construction (Jesper Grodal)
Thur Nov 3: Quillen's calculation of the K-theory of a finite field (Jesper Grodal)
Tues Nov 8: Proofs of Quillen's Thm A+B and related theorems (Mike Shulman)
Thur Nov 10: Low dimensional K-groups, arithmetic, and Milnor K-theory (Masoud Kamgarpour)
Tues Nov 15: K-theory via the +, Q and S. construction; comparison (following Weibels prebook) I (Bert Guillou)
Thur Nov 17: K-theory via the +, Q and S. construction II (Bert Guillou)
Tues Nov 22: K-theory via the +, Q and S. construction III (Bert Guillou)
Thur Nov 24: No seminar (Thanksgiving).
Tues Nov 29: Fundamental theorems in alg K-theory (additivity, localization, devisage, follow\ing McCarthy+Staffeldt) I (Justin Noel)
Thur Dec 1: Fundamental theorems in alg K-theory II (Justin Noel)

COMING THIS WINTER (?):
Equivariant homotopy theory (equivariant theory and diagram categories, smith theory, Atiyah segal completion thm, Segal conj, Sullivan conj, homotopy fixed point methods, transformation groups (Adem-Davis survey), classifying spaces, orbifolds and twisted K-theory, cohomology of finite, arithmetic, and S-arithmetic groups, and the relationship to Friedlander-Milnor and unstable Quillen-Lichtenbaum conjectures, finite loop spaces are manifolds) (Jesper Grodal, Johann Leida,...)

Structured ring spectra (following outline of Arbeitsgemeinshaft, also: Morita theory survey of Schwede and duality paper of Dwyer-Greenlees-Iyengar) (Nathalie Wahl, Peter May, etc...)

Homological stability theorems (for symmetric groups, braid groups, linear groups, etc...) (Nathalie Wahl etc.)

Calculus of functors (going through Calculus III by Goodwillie and survey by Kuhn)

PRACTICE TALKS:
Everyone should give at least one practice talk before their real talk, and giving as well as attending practice talks is an integral part of the course. Niles Johnson (niles at math...) is in charge of arranging the time and place for the practice talks. Contact him for more information.

PREVIOUS QUARTERS:
Fall 2004 , Winter 2005 , Spring 2005 .

OTHER STUFF: For inspiration here's a link to the student topology seminar "babytop" at MIT , the student topology reading seminar "juvitop" at MIT and the Kan seminar .