School of Mathematics

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

Yuanzhi Li
Princeton University
March 19, 2018

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 Weyl law for algebraic tori

Ian Petrow
ETH Zurich
March 13, 2018

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.

Abstract Convexity, Weak Epsilon-Nets, and Radon Number

Shay Moran
University of California, San Diego; Member, School of Mathematics
March 13, 2018

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.
 

Higher ribbon graphs

David Nadler
University of California, Berkeley
March 12, 2018

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.

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