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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 ) .

Endoscopy and cohomology growth on unitary groups

Simon Marshall
University of Wisconsin; Member, School of Mathematics
March 9, 2018
Abstract: One of the principles of the endoscopic classification is that if an automorphic representation of a classical group is non-tempered at any place, then it should arise as a transfer from an endoscopic subgroup. One also knows that any representation of a unitary group that contributes to the cohomology of the associated symmetric space outside of middle degree must be non-tempered at infinity. By combining these two ideas, I will derive conjecturally sharp upper bounds for the growth of Betti numbers in congruence towers of complex hyperbolic manifolds.

Ax-Schanuel for Shimura Varieties

Jacob Tsimerman
University of Toronto
March 9, 2018
Abstract: (joint with N.Mok, J.Pila) Shimura varieties (S) are uniformized by symmetric spaces (H), and the uniformization map Pi:H --> S is quite transcendental. Understanding the interaction of this map with the two algebraic structures is of particular interest in arithmetic, as it is a necessary ingredient for the modern approaches to the Andre-Oort and Zilber-Pink conjectures, as well

The Presend State of the Jacquet-Rallis trace formula

Pierre-Henri Chaudouard
IMJ PRG
March 9, 2018
Abstract: The Jacquet-Rallis trace formula is a powerful tool to
study periods of automorphic forms of unitary groups that show
up in Ichino-Ikeda conjecture. In this talk, I will report on the
present state of the Jacquet-Rallis trace formula. Then I will
discuss the problem of the spectral expansion. (joint work with Michal
Zydor).

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 $(GL_1 , GL_N )$.

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

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.

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.

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.