# School of Mathematics

## Bounds for character twists of L-functions

## The Weyl law for algebraic tori

A basic but difficult question in the analytic theory of automorphic forms is: given a reductive group G and a representation r of its L-group, how many automorphic representations of bounded analytic conductor are there? In this talk I will present an answer to this question in the case that G is a torus over a number field.

## Math for underprivileged high school kids

We will hear from two passionate creators of successful mentoring programs in math for high school kids in educationally challenged environments. They will give back-to-back talks about their experiences and educational insights.

## Higher ribbon graphs

Ribbon graphs capture the topology of open Riemann surfaces in an elementary combinatorial form. One can hope this is the first step toward a general theory for open symplectic manifolds such as Stein manifolds. We will discuss progress toward such a higher dimensional theory (joint work with Alvarez-Gavela, Eliashberg, and Starkston), and in particular, what kind of topological spaces might generalize graphs. We will also discuss applications to the calculation of symplectic invariants.

## Endoscopy and cohomology growth on unitary groups

## Nodal domains for Maass forms

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 ) .

## The Presend State of the Jacquet-Rallis trace formula

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).

## Ax-Schanuel for Shimura Varieties

## Special cycles on simple Shimura varieties

simple algebras with an involution of second kind. We study some local questions arising from the relative trace formula approach.