School of Mathematics

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.

Nonlinear dimensionality reduction for faster kernel methods in machine learning.

Christopher Musco
Massachusetts Institute of Technology
February 12, 2018

The Random Fourier Features (RFF) method (Rahimi, Recht, NIPS 2007) is one of the most practically successful techniques for accelerating computationally expensive nonlinear kernel learning methods. By quickly computing a low-rank approximation for any shift-invariant kernel matrix, RFF can serve as a preprocessing step to generically accelerate algorithms for kernel ridge regression, kernel clustering, kernel SVMs, and other benchmark data analysis tools.
 

Outlier-Robust Estimation via Sum-of-Squares

Pravesh Kothari
February 6, 2018

We develop efficient algorithms for estimating low-degree moments of unknown distributions in the presence of adversarial outliers. The guarantees of our algorithms improve in many cases significantly over the best previous ones, obtained in recent works. We also show that the guarantees of our algorithms match information-theoretic lower-bounds for the class of distributions we consider. These better guarantees allow us to give improved algorithms for independent component analysis and learning mixtures of Gaussians in the presence of outliers.