University of Chicago Number Theory Seminar

Winter 2010: Friday 3:30-5:00pm, room E207

This is the home page of Number Theory Seminar at University of Chicago. To get on (or off) the seminar mailing list, please contact Sug Woo Shin or Liang Xiao.

Click here to see the location of Eckhart Hall, and here for driving directions to University of Chicago.

One may also want to check out Geometric Langlands Seminar and Algebraic Geometry Seminar.

Click here to see the schedule of previous quarters: Winter 2010 / Spring 2010 / Fall 2010 / Winter 2011 / Spring 2011 / Fall 2011 / Winter 2012 / Spring 2012 / Fall 2012 / Winter 2013 / Spring 2013 / Fall 2013 / Winter 2014

When an abstract is available, click on "Show Abstract" to expand the abstract, or click on "Hide Abstract" to hide it.





January 15

Takashi Suzuki

Some remarks on local class field theory of Serre and Hazewinkel
We briefly review local class field theory of Serre and Hazewinkel, and give a new approach for this theory. In the case of characteristic zero, we also explain a D-module version of the theory. Two-dimensional local class field theory is discussed in this framework.

January 22

Sug Woo Shin

Cohomology of locally symmetric hermitian spaces
In this expository talk (meant to be educational even for the speaker himself), we introduce the idea of Faltings, Harris and others to study the cohomology of locally symmetric hermitian spaces. One motivating question is how to
1) obtain a refinement of the Hodge decomposition of the singular cohomology (with twisted coefficients) and
2) relate each piece to automorphic forms or representations.

January 29

Keerthi Madapusi

Integral Canonical Models of Shimura varieties of Hodge type (after Faltings and Kisin)
This will be an introduction to the techniques introduced by Faltings, Kisin et al. to construct integral canonical models of Shimura varieties of Hodge type. There will be inputs from the theory of Hodge cycles on abelian varieties, the deformation theory of p-divisible groups, and integral p-adic Hodge theory, and attempts will be made to describe all of them precisely. This talk again will be of an expository nature.

February 5

Tasho Kaletha

Depth-zero local Langlands correspondence and endoscopic transfer
In this talk, we will motivate the problem that the theory of endoscopy adresses, and then formulate the precise statement of the endoscopic character identities, after recalling the necessary notions from the local Langlands correspondence. We will then discuss their proof for the depth-zero supercuspidal L-packets recently constructed by DeBacker-Reeder. The main technical tool involved is Waldspurger's work on endoscopy for p-adic Lie algebras, which ultimately rests on the fundamental lemma. more

February 12

No Meeting

College Break

February 19

Abhik Ganguli

On the mod p reduction of certain two dimensional crystalline representations
In this talk we will consider the problem of determining explicitly the mod p reduction (up to semisimplification) of a two dimensional irreducible Galois representation which is crystalline at p. In particular, we assume that the Galois representation comes from a certain quaternionic form. We will determine the reduction with no restrictions on the Hodge-Tate weights, confining the range of the p-adic valuation of the crystalline slope of our Galois representation. Our method relies on a certain weight lowering technique. We show that the mod p Galois representation comes from an eigenform of sufficiently low weight, yielding certain congruences of modular forms on our way.

February 26

Ben McReynolds

Bertrand's postulate and subgroup growth
I will discuss a few generalizations of Bertrand's postulate on primes to finitely generated linear groups recently done jointly with Khalid Bou-Rabee. This topic has connections to subgroup growth and the L1-norm of certain functions on profinite groups called divisibility functions. The talk will be accessible to undergraduate math majors. Indeed, results like the Chinese Remainder Theorem, the Ratio Test, and L'Hopital's rule are some of our main tools.

March 5

Florian Herzig

The classification of irreducible mod p representations of a p-adic GLn
Let F be a finite extension of the p-adic numbers. We describe the classification of irreducible admissible smooth representations of GLn(F) over an algebraically closed field of characteristic p, in terms of "supersingular" representations. This generalizes results of Barthel-Livne for n = 2. Our motivation is the hypothetical mod p Langlands correspondence for GLn, which is supposed to relate smooth mod p representations of GLn(F) to n-dimensional mod p Galois representations.

March 12

Liang Xiao

The slope filtration theorem
This will be an expository talk on Kedlaya's slope filtration theorem. Instead of talking about applications, we focus on the actual proof of the theorem.

March 19

Skip Garibaldi (Emory University)

There is no "Theory of Everything" inside E8
Note the different time. (Special geometry/number theory seminar)
The "Exceptionally Simple Theory of Everything" has been the subject of articles in The New Yorker (7/21/08), Le Monde (11/20/07), the Financial Times (4/25/09), The Telegraph (11/10/09), an invited talk at TED (2/08), etc. Despite positive descriptions of the theory in the popular press, it doesn't work. I'll explain a little of the theory, some reasons why it doesn't work, and a theorem (joint with physicist Jacques Distler) about Lie groups that says that no similar theory can work. This talk should be accessible to all graduate students in mathematics.

March 19

Atsushi Ichino (Osaka City Univ.)

Formal degrees and local theta correspondence
The formal degree conjecture, which was formulated with K. Hiraga and T. Ikeda, relates a certain representation-theoretic invariant to an arithmetic invariant. It seems hard to prove it but possible to test its functoriality property. We discuss the case of local theta correspondence. If time permits, we also discuss a relation with the Siegel-Weil formula. This is joint work with Wee Teck Gan.

This page is maintained by Liang Xiao; it was shamelessly copied from Kiran Kedlaya, which in turn was shamelessly copied from Jason Starr's page, which in turn was shamelessly copied from Ravi Vakil's page, which in turn was shamelessly copied from Pasha Belorousski's page at the University of Michigan. For more sites with a similar pedigree, see Michael Thaddeus's list or Jim Bryan's list.