School of Mathematics

Two questions of Landis and their applications

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.

Harmonic maps into singular spaces

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

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).