## Ramanujan complexes and golden gates in PU(3).

Shai Evra

Member, School of Mathematics

January 9, 2019

In their seminal works from the 80's, Lubotzky, Phillips and Sarnak proved the following two results:

Shai Evra

Member, School of Mathematics

January 9, 2019

Franco Vargas Pallete

University of California, Berkeley; Member, School of Mathematics

December 18, 2018

Renormalized volume (and more generally W-volume) is a geometric quantity found by volume regularization. In this talk I'll describe its properties for hyperbolic 3-manifolds, as well as discuss techniques to prove optimality results.

Sobhan Seyfaddini

ENS Paris

December 17, 2018

Hamiltonian homeomorphisms are those homeomorphisms of a symplectic manifold which can be written as uniform limits of Hamiltonian diffeomorphisms. One difficulty in studying Hamiltonian homeomorphisms (particularly in dimensions greater than two) has been that we possess fewer tools for studying them. For example, (filtered) Floer homology, which has been a very effective tool for studying Hamiltonian diffeomorphisms, is not well-defined for homeomorphisms.

Eugenia Malinnikova

NTNU; von Neumann Fellow, School of Mathematics

December 14, 2018

We discuss two old questions of Landis concerning behavior of solutions of second order elliptic equations. The first one is on

propagation of smallness for solutions from sets of positive measure,

we answer this question and as a corollary prove an estimate for eigenfunctions of Laplace operator conjectured by Donnelly and Fefferman. The second question is on the decay rate of solutions of the Schrödinger equation, we give the answer in dimension two. The talk is based on joint work with A. Logunov, N. Nadirashvili, and F. Nazarov.

propagation of smallness for solutions from sets of positive measure,

we answer this question and as a corollary prove an estimate for eigenfunctions of Laplace operator conjectured by Donnelly and Fefferman. The second question is on the decay rate of solutions of the Schrödinger equation, we give the answer in dimension two. The talk is based on joint work with A. Logunov, N. Nadirashvili, and F. Nazarov.

Michael Walter

University of Amsterdam

December 11, 2018

Andre Neves

University of Chicago; Member, School of Mathematics

December 11, 2018

I will outline the proof of density of minimal hypersurfaces (Irie-Marques-Neves) and equidistribution of minimal hypersurfaces (Marques-Neves-Song).

Brian Freidin

Brown University; Visitor, School of Mathematics

December 11, 2018

In the 90's, Gromov and Schoen introduced the theory of

harmonic maps into singular spaces, in particular Euclidean buildings,

in order to understand p-adic superrigidity. The study was quickly

generalized in a number of directions by a number of authors. This

talk will focus on the work initiated by Korevaar and Schoen on

harmonic maps into metric spaces with curvature bounded above in the

sense of Alexandrov. I will describe the variational characterization

harmonic maps into singular spaces, in particular Euclidean buildings,

in order to understand p-adic superrigidity. The study was quickly

generalized in a number of directions by a number of authors. This

talk will focus on the work initiated by Korevaar and Schoen on

harmonic maps into metric spaces with curvature bounded above in the

sense of Alexandrov. I will describe the variational characterization

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.

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.

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.