# School of Mathematics

## Operator Scaling via Geodesically Convex Optimization, Invariant Theory and Polynomial Identity Testing

We propose a new second-order method for geodesically convex optimization on the natural hyperbolic metric over positive definite matrices. We apply it to solve the operator scaling problem in time polynomial in the input size and logarithmic in the error. This is an exponential improvement over previous algorithms which were analyzed in the usual Euclidean, “commutative” metric (for which the above problem is not convex).

## The hidden landscape of localization

## Abstract Convexity, Weak Epsilon-Nets, and Radon Number

Let F be a family of subsets over a domain X that is closed under taking intersections. Such structures are abundant in various fields of mathematics such as topology, algebra, analysis, and more. In this talk we will view these objects through the lens of *convexity*.

We will focus on an abstraction of the notion of *weak epsilon nets*:

given a distribution on the domain X and epsilon>0,

a weak epsilon net for F is a set of points that intersects any set in F with measure at least epsilon.

## Period mappings are definable in the o-minimal structure $\mathbb{R}_{an,exp}$

## 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.

## 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).