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Local to global relations of periods (continued)

Erez Lapid
Weizmann Institute of Science; Member, School of Mathematics
February 27, 2018

Rankin-Selberg integrals provide factorization of certain period integrals into local counterparts. Other, more elusive, periods can be studied in principle by the relative trace formula and other methods.
 
Following Waldspurger, Ichino-Ikeda formulated a local to global conjecture about the Gross-Prasad periods. A more general setup was subsequently considered by Sakellaridis-Venkatesh.
 
I will discuss some of these principles as well as a result on Whittaker coefficients joint with Zhengyu Mao.

Representations of p-adic groups

Jessica Fintzen
University of Michigan
February 26, 2018

I will survey what is known about the construction of (the building blocks of) representations of p-adic groups, mention recent developments, and explain some of the concepts underlying all constructions. In particular, I will introduce filtrations of p-adic groups and indicate some of their remarkable properties.

A Tight Bound for Hypergraph Regularity

Guy Moshkovitz
Harvard University
February 26, 2018

The hypergraph regularity lemma — the extension of Szemeredi's graph regularity lemma to the setting of k-graphs — is one of the most celebrated combinatorial results obtained in the past decade. By now there are various (very different) proofs of this lemma, obtained by Gowers, Rodl et al. and Tao. Unfortunately, what all these proofs have in common is that they yield partitions whose order is given by the k-th Ackermann function.
 

Local to global relations of periods

Erez Lapid
Weizmann Institute of Science; Member, School of Mathematics
February 20, 2018

Rankin-Selberg integrals provide factorization of certain period integrals into local counterparts. Other, more elusive, periods can be studied in principle by the relative trace formula and other methods.
 
Following Waldspurger, Ichino-Ikeda formulated a local to global conjecture about the Gross-Prasad periods. A more general setup was subsequently considered by Sakellaridis-Venkatesh.
 
I will discuss some of these principles as well as a result on Whittaker coefficients joint with Zhengyu Mao.

Some closure results for polynomial factorization

Mrinal Kumar
Harvard University
February 20, 2018

In a sequence of extremely fundamental results in the 80's, Kaltofen showed that any factor of n-variate polynomial with degree and arithmetic circuit size poly(n) has an arithmetic circuit of size poly(n). In other words, the complexity class VP is closed under taking factors.
 
A very basic question in this context is to understand if other natural classes of multivariate polynomials, for instance, arithmetic formulas, algebraic branching programs, bounded depth arithmetic circuits or the class VNP, are closed under taking factors.
 

On the long-term dynamics of nonlinear dispersive evolution equations

Wilhelm Schlag
University of Chicago Visiting Professor, School of Mathematics
February 14, 2018

We will give an overview of some of the developments in recent years dealing with the description of asymptotic states of solutions to semilinear evolution equations ("soliton resolution conjecture").
 
New results will be presented on damped subcritical Klein-Gordon equations, joint with Nicolas Burq and Genvieve Raugel.

Abstract homomorphisms of algebraic groups and applications

Igor Rapinchuk
Michigan State University
February 13, 2018

I will discuss several results on abstract homomorphisms between the groups of rational points of algebraic groups. The main focus will be on a conjecture of Borel and Tits formulated in their landmark 1973 paper.
 
Our results settle this conjecture in several cases; the proofs make use of the notion of an algebraic ring. I will mention several applications to character varieties of finitely generated groups and representations of some non-arithmetic groups.