February 1  Dimitri Wyss (Sorbonne University Paris 6)

padic integration and geometric stabilization
I will explain a new proof of the geometric stabilization theorem for Hitchin fibers, a key ingredients in Ngô's proof of the fundamental lemma. Our approach relies on ideas of DenefLoeser and Batyrev on padic integration and Langlands duality for generic Hitchin fibers. This is joint work with Michael Groechenig and Paul Ziegler.

February 8  Hang Xue (Arizona)

Epsilon dichotomy for linear models
SaitoTunnell theorem is a local version of Waldspurger's formula, relating the existence of E^\times invariant linear forms on representation of GL_2 to local root numbers. I present a generalization of this which relates the existence of GL_n(E) invariant linear forms on GL_2n(F) to local root numbers. The proof relies on a "relative version" of HarishChandra' theory of local character expansions.

February 15  Xinyi Yuan (Berkeley)

Weak Lefschetz theorems for Brauer groups
In this talk, we introduce some Lefschetztype theorems for Brauer groups of hyperplane sections of smooth projective varieties. This is more or less known when the dimension of the hyperplane section is at least 3, but we will also introduce a version which lowers the dimension from 3 to 2. As a consequence, we reduce the Tate conjecture for divisors on smooth projective varieties from general dimensions to dimension 2, and thus proves a results of Morrow by a different method.

February 22  Baiying Liu (Purdue)

On automorphic descent from GL(7) to G2
In this talk, I will introduce the functorial descent from cuspidal automorphic representations \pi of GL7(A) with L^S(s, \pi, \wedge^3) having a pole at s=1 to the split exceptional group G2(A), using Fourier coefficients associated to two nilpotent orbits of E7. We show that one descent module is generic, and under mild assumptions on the unramified components of \pi, it is cuspidal and having \pi as a weak functorial lift of each irreducible summand. However, we show that the other descent module supports not only the nondegenerate Whittaker integeral on G2(A), but also every degenerate Whittaker integral. Thus it is generic, but not cuspidal. This is a new phenomenon, compared to the theory of functorial descent for classical and GSpin groups. This work is joint with Joseph Hundley.

March 1  Rong Zhou (IAS)

Motivic cohomology of quaternionic Shimura varieties and level raising
For a smooth scheme of finite type over a field its motivic cohomology groups generalize the usual Chow groups and are an important algebraic invariant. In this talk we will explain how, for the special fibers of certain quaternionic Shimura varieties, its motivic cohomology can encode very rich arithmetic information. More precisely we will show that the cycle class map from motivic cohomology to étale cohomology gives a geometric realization of level raising between Hilbert modular forms. The main ingredient for this construction is a form of Ihara's Lemma for Shimura surfaces which we prove by generalizing a method of DiamondTaylor.

March 8  Rahul Krishna (Northwestern)

The semiSchrödinger model of an exceptional representation of $\widetilde{GL}_{2q}$
In 1984, Kazhdan and Patterson constructed what are now called the exceptional representations of metaplectic $r$fold covers of $GL_n$. These representations, which include the Weil representation when $r=n=2$, have seen numerous applications; most spectacularly, when $r=2$ and $n$ is arbitrary, they appear in the integral representation of the symmetric square $L$ function on $GL_n$. Unfortunately they are somewhat difficult to work with in practice, even for $r=2$ and $n>2$. In this talk, I will explain how, for $r=2$ and $n=2q$, these representations can be described in terms of a model space akin to the Schrödinger model of the Weil representation. I will also explain some of my motivations for this problem, and some possible applications of this result.

March 29  Luis Garcia (Toronto)

Transgression of the Euler class and Eisenstein cohomology
I will discuss a new construction of Eisenstein cohomology
classes on GL_N first introduced by Nori and Sczech. In our approach the
corresponding cocycles appear as regularised theta lifts for the dual pair
(GL_1,GL_N). This suggests interesting generalisations of this
construction by considering more general pairs (GL_k, GL_N), and I will
present some calculations regarding this generalisation. Joint work (in
progress) with Nicolas Bergeron, Pierre Charollois and Akshay Venkatesh.

April 5  Guangyu Zhu (Yale)

The Galois group of the category of mixed HodgeTate structures
The category of mixed HodgeTate structures over Q is a mixed Tate category of homological dimension one. By Tannakian formalism, it is equivalent to the category of graded comodules of a commutative graded Hopf algebra. In my recent joint work with A. Goncharov, we give a canonical description A (C) of the Hopf algebra. Such construction can be generalized to A (R) for any dgalgebra R with a Tate line.

April 19  Wanlin Li (WisconsinMadison)

Vanishing of Hyperelliptic Lfunctions at the Central Point
We study the number of quadratic Dirichlet Lfunctions over the rational function field which vanish at the central point s=1/2. In the first half of my talk, I will give a lower bound on the number of such characters through a geometric interpretation. This is in contrast with the situation over the rational numbers, where a conjecture of Chowla predicts there should be no such Lfunctions. In the second half of the talk, I will discuss joint work with Ellenberg and Shusterman proving as the size of the constant field grows to infinity, the set of Lfunctions vanishing at the central point has 0 density.

April 26  Weizhe Zheng (MCM/Princeton)

Compatible systems along the boundary
A theorem of Deligne says that compatible systems of ladic sheaves on a smooth curve over a finite field are compatible along the boundary. I will present an extension of Deligne's theorem to schemes of finite type over the ring of integers of a local field. This has applications to lindependence of ladic cohomology of varieties over Henselian valuation fields, possibly of higher rank. This is joint work with Qing Lu.

May 3  Charlotte Chan (Princeton)

Flag varieties and representations of padic groups
DeligneLusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive
groups over finite fields. In this talk, we discuss recent progress
towards geometric realizations of representations of padic groups in
three arcs: affine flag varieties, semiinfinite flag varieties, and a
finitering variant arising from the MoyPrasad filtration of
parahoric subgroups. This is joint work with Alexander Ivanov.

May 10  Raphaël BeuzartPlessis (AixMarseille University)

Spectral decomposition of certain symmetric spaces and
the IchinoIkeda conjecture
The IchinoIkeda conjecture expresses the central value of certain
automorphic Lfunctions in terms of the square of an automorphic period. It is
a higher rank generalization of a wellknown theorem of Waldspurger on toric
periods and is closely related to the global GanGrossPrasad conjecture. In
this talk, I am going to explain how explicit Plancherel decompositions of
certain symmetric spaces allow to complement previous work of W.Zhang to
establish this conjecture for unitary groups under a mild local assumption.
Time permitting, I will also explain another application of these Plancherel
decompositions to a conjecture of HiragaIchinoIkeda relating formal degrees
of discrete series to adjoint gamma factors.
