School of Mathematics

A matrix expander Chernoff bound

Ankit Garg
Microsoft Research
December 10, 2018
Chernoff-type bounds study concentration of sums of independent random variables and are extremely useful in various settings. In many settings, the random variables may not be completely independent but only have limited independence. One such setting, which turns out to be useful in derandomization and theoretical computer science, in general, involves random walks on expanders. I will talk about a Chernoff-type bound for sums of matrix-valued random variables sampled via a random walk on an expander.

Morrey Saces and Regularity for Yang-Mills Higgs Equations

Karen Uhlenbeck
School of Mathematics
December 7, 2018
We start with background on regularity theory for the equations of gauge theory. Morrey spaces arise naturally from monotinicity theorems in dimensions greater than 4. Our main technical result is that functions in a Morrey space which satisfy an elliptic inequality off a singular set of Hausdorf codimension 4 can be bounded in a much better Morrey space in the interior.

Slopes in eigenvarieties for definite unitary groups

Lynnelle Ye
Harvard University
December 6, 2018
The study of eigenvarieties began with Coleman and Mazur, who constructed the first eigencurve, a rigid analytic space parametrizing $p$-adic modular Hecke eigenforms. Since then various authors have constructed eigenvarieties for automorphic forms on many other groups. We will give bounds on the eigenvalues of the $U_p$ Hecke operator appearing in Chenevier's eigenvarieties for definite unitary groups. These bounds generalize ones of Liu-Wan-Xiao for dimension $2$, which they used to prove a conjecture of Coleman-Mazur-Buzzard-Kilford in that setting, to all dimensions.

Global results related to scalar curvature and isoperimetry

Otis Chodosh
Princeton University; Veblen Research Instructor, School of Mathematics
December 4, 2018
I will first survey some recent progress on global problems related to scalar curvature and area/volume, focusing in particular on scale breaking phenomena in such problems. I will then discuss the role of the Hawking mass in the resolution of this scale-breaking issue for the stable CMC uniqueness problem in asymptotically Schwarzschild manifolds (joint work with M. Eichmair) and possibly mention some features of the isoperimetric problem in asymptotically Schwarzschild-anti-de Sitter manifolds (joint work with M. Eichmair, Y. Shi, J. Zhu).

Recent Progress on Zimmer's Conjecture

David Fisher
Indiana University, Bloomington; Member, School of Mathematics
December 3, 2018
Lattices in higher rank simple Lie groups are known to be extremely rigid. Examples of this are Margulis' superrigidity theorem, which shows they have very few linear represenations, and Margulis' arithmeticity theorem, which shows they are all constructed via number theory. Motivated by these and other results, in 1983 Zimmer made a number of conjectures about actions of these groups on compact manifolds and in a recent breakthrough with Brown and Hurtado we have proven many of them.

Mean action of periodic orbits of area-preserving annulus diffeomorphisms

Morgan Weiler
University of California, Berkeley
December 3, 2018
An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit.

Branched conformal structures and the Dyson superprocess

Govind Menon
Brown University; Member, School of Mathematics
November 30, 2018

In the early 1920s, Loewner introduced a constructive approach to the Riemann mapping theorem that realized a conformal mapping as the solution to a differential equation. Roughly, the “input” to Loewner’s differential equation is a driving measure and the “output” is a family of nested, conformally equivalent domains. This theory was revitalized in the late 1990s by Schramm. The Schramm-Loewner evolution (SLE) is a stochastic family of slit mappings driven by Loewner’s equation when the driving measure is an atom executing Brownian motion.

The Lucky Logarithmic Derivative

Will Sawin
Columbia University
November 29, 2018
We study the function field analogue of a classical problem in analytic number theory on the sums of the generalized divisor function in short intervals, in the limit as the degrees of the polynomials go to infinity. As a corollary, we calculate arbitrarily many moments of a certain family of L-functions, in the limit as the conductor goes to infinity. This is done by showing a cohomology vanishing result using a general bound due to Katz and some elementary calculations with polynomials.