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.
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.
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.
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.
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