Nodal domains for Maass forms

Peter Sarnak
Professor, School of Mathematics
March 9, 2018
Abstract: A highly excited Maass form on a hyperbolic surface
is expected to behave like a random monochromatic wave .
We will discuss this in connection with the question of the nodal
domains of such forms on arithmetic hyperbolic surfaces with a reflection symmetry .
( Joint work with A.Ghosh and A.Reznikov we will also discuss a recent result of
J.Jang and J.Jung ) .

Special cycles on simple Shimura varieties

Wei Zhang
Massachusetts Institute of Technology
March 8, 2018
Abstract: This is a work in progress inspired by the the arithmetic Gan--Gross--Prasad conjecture where one is interested in the arithmetic diagonal cycle on the product of two Shimura varieties. We study special cycles on simple Shimura varieties attached to central
simple algebras with an involution of second kind. We study some local questions arising from the relative trace formula approach.

Admissible heigh pairings of algebraic cycles

Shouwu Zhang
Princeton University
March 8, 2018
Abstract: For a smooth and projective variety X over a global field of dimension n with an adelic polarization, we propose canonical local and global height pairings for two cycles Y, Z of pure dimension p, q satisfying $p+q=n-1$. We will give some explicit arichmedean local pairings by writing down explicit formula for the diagonal green current for some Shimura varieties.

Arithmetic theta series

Stephan Kudla
University of Toronto
March 8, 2018
Abstract: In recent joint work with Jan Bruinier, Ben Howard, Michael Rapoport and Tonghai Yang,
we proved that a certain generating series for the classes of arithmetic divisors on a regular integral model M of a Shimura variety
for a unitary group of signature (n-1,1) for an imaginary quadratic field is a modular form of weight n valued in the
first arithmetic Chow group of M. I will discuss how this is proved, highlighting the main steps.
Key ingredients include information about the divisors of Borcherds forms on the integral model

Euler classes transgressions and Eistenstein cohomology of GL(N)

Nicolas Bergeron
IMJ PRG
March 8, 2018
Abstract: In work-in-progress with Pierre Charollois, Luis Garcia and Akshay Venkatesh we give a new construction of some Eisenstein classes for $GL_N (Z)$ that were first considered by Nori and Sczech. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of $SL_N$ (Z)-vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a regularized theta lift for the reductive dual pair $(GL_1 , GL_N )$.

Restriction problem for non-generic representation of Arthur type

Wee Teck Gan
National University of Singapore
March 6, 2018
Abstract: The Gross-Prasad conjecture considers a branching problem for generic Arthur packets of classical groups. In this talk, we will describe progress towards extending this conjecture to nongeneric Arthur packets (this is joint work with Gross and Prasad). For GL(n), we describe some recent progress towards this conjecture by Max Gurevich.

Relative character asymptotics and applications

Paul Nelson
ETH Zurich
March 6, 2018
Abstract: Relative (or spherical) characters describe the restriction of a representation to a subgroup. They arise naturally in the study of periods of automorphic forms, e.g., in the setting of conjectures of Gan-Gross-Prasad and Ichino-Ikeda. I will discuss the problem of their asymptotic estimation, emphasizing some known results, open problems and applications.

The Plancherel formula for L^2(GL_n(F)\GL_n(E)) and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups

Raphael Beuzart-Plessis
CNRS
March 6, 2018
Abstract : Let $E/F$ be a quadratic extension of local fields of characteristic zero. In this talk, I will explain two ways to compute the Plancherel decomposition of $L^2(GL_n(F)\backslash GL_n(E))$. In both cases, the result involves the image of base change from unitary groups to $GL_n(E)$ and is in accordance with a general conjecture of Sakellaridis-Venkatesh on the spectral decomposition of spherical varieties. We will also give applications of our formulas to the so-called Ichino-Ikeda and formal degree conjectures for unitary groups.