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Fri Sep 14

Lie Theory Seminar

2:30pm - Vincent Hall 313
D-modules, perverse sheaves, and the Riemann-Hilbert correspondence: an overview
Kai-Wen Lan, University of Minnesota
Fri Oct 19

Lie Theory Seminar

4:30pm - Vincent Hall 313
On Several Classical Number Theory Problems
Tianxin Cai, Zhejiang University and University of Iowa
Fri Nov 09

Lie Theory Seminar

3:35pm - Vincent Hall 313
Motivic cohomology of Shimura varieties and level raising
Rong Zhou, Institute for Advanced Study

For a finite type scheme over a field, its motivic cohomology groups were defined by Voevodsky and are an important algebraic invariant. However, the properties of these groups are not well understood, and it is a difficult problem to exhibit explicit classes in motivic cohomology. We will construct such classes in the special fiber of Hilbert modular varieties by using the geometry of the supersingular locus. The construction is related to a geometric realization of the Jacquet-Langlands correspondence, as well as to level raising for Hilbert modular forms. A key ingredient is a form of Ihara's Lemma for compact quaternionic Shimura surfaces.

Fri Dec 07

Lie Theory Seminar

3:35pm - Vincent Hall 313
Finiteness of Frobenius traces of a de Rham local system
Koji Shimizu, University of California, Berkeley

Every smooth projective variety over a number field yields a Galois representation via etale cohomology, and the Weil conjecture tells that its Frobenius traces are integers. Fontaine and Mazur conjectured that Galois representations satisfying a local condition (de Rham) arise from geometry and hence have a similar finiteness property. In this talk, I will focus on de Rham local systems on algebraic varieties and explain a finiteness of Frobenius traces follows from the Fontaine-Mazur conjecture for Galois representations and the generalized Riemann Hypothesis

Fri Feb 15

Lie Theory Seminar

3:35pm - Vincent Hall 313
Lie Theory Seminar
Lue Pan,, University of Chicago
Wed Apr 24

Lie Theory Seminar

1:25pm - Vincent Hall 301
A new proof of the Jacquet-Rallis fundamental lemma
Professor Raphael Beuzart-Plessis, CNRS and Columbia University

The Jacquet-Rallis fundamental lemma is a local identity between (relative) orbital integrals which originates from the relative trace formula approach to the Gan-Gross-Prasad conjecture for unitary groups and is a crucial ingredient in the recent results of W. Zhang on this conjecture. It was established soon after its formulation by Z. Yun in positive characteristic using the same geometric ideas as in Ngô's proof of the endoscopic fundamental lemma and transferred to characteristic 0 by J. Gordon by model-theoretic techniques. In this talk, I will present an alternative proof of this fundamental lemma in characteristic zero which is purely local and based on harmonic analytic tools. We note that a third proof of this fundamental lemma has been recently proposed by W. Zhang through a similar although global argument.