## Deep Learning: A Scientific Perspective

Nadav Cohen

Member, School of Mathematics

January 11, 2019

Nadav Cohen

Member, School of Mathematics

January 11, 2019

Naser Talebi Zadeh Sardari

University of Wisconsin Madison

January 9, 2019

Motivated by questions in computer science, we consider the problem of approximating local points (real or p-adic points) on the unit sphere S^d optimally by the projection of the integral points lying on R*S^d, where R^2 is an integer. We present our numerical results which show the diophantine exponent of local point on the sphere is inside the interval [1, 2-2/d]. By using the Kloosterman's circle method, we show that the diophantine exponent is less than 2-2/d for every d>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:

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.

Eric Michaud

Directeur d’Études at the Ecole des Hautes Études en Sciences Sociales and former Member in the School of Historical Studies.

December 12, 2018

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