Date

Speaker

Topic

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 Dmodule version of the theory.
Twodimensional 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 pdivisible
groups, and integral padic 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

Depthzero 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 depthzero supercuspidal Lpackets recently constructed by DeBackerReeder. The main technical tool involved is Waldspurger's work on endoscopy for padic 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 HodgeTate weights, confining the range of the
padic 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
BouRabee. This topic has connections to subgroup growth and the L^{1}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 (Northwestern)

The classification of irreducible mod p
representations of a padic GL_{n}
Let F be a finite extension of the padic numbers. We describe
the classification of irreducible admissible smooth
representations of GL_{n}(F) over an algebraically closed field of
characteristic p, in terms of "supersingular" representations.
This generalizes results of BarthelLivne for n = 2. Our
motivation is the hypothetical mod p Langlands
correspondence for GL_{n}, which is supposed to relate smooth
mod p representations of GL_{n}(F) to ndimensional 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 23:15pm

Skip Garibaldi (Emory University)

There is no "Theory of Everything" inside E_{8}
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 representationtheoretic
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 SiegelWeil formula. This is joint work with Wee
Teck Gan.
