(Non)uniqueness questions in mean curvature flow

Lu Wang
University of Wisconsin–Madison; Member, School of Mathematics
January 22, 2019

Mean curvature flow is the negative gradient flow of the
volume functional which decreases the volume of (hyper)surfaces in the
steepest way. Starting from any closed surface, the flow exists
uniquely for a short period of time, but always develops singularities
in finite time. In this talk, we discuss some non-uniqueness problems
of the mean curvature flow passing through singularities. The talk is
mainly prepared for non-specialists of geometric flows.

Regularity of weakly stable codimension 1 CMC varifolds

Neshan Wickramasekera
University of Cambridge; Member, School of Mathematics
January 15, 2019
The lecture will discuss recent joint work with C. Bellettini and O. Chodosh. The work taken together establishes sharp regularity conclusions, compactness and curvature estimates for any family of codimension 1 integral $n$-varifolds satisfying: (i) locally uniform mass and $L^{p}$ mean curvature bounds for some $p > n;$ (ii) two structural conditions and (iii) two variational hypotheses on the orientable regular parts, namely, stationarity and (weak) stability with respect to the area functional for volume preserving deformations (supported on the regular parts).

Distribution of the integral points on quadrics

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.

Barcodes and $C^0$ symplectic topology

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.

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.