3:45PM: Padic Gaussians and their tropicalizationYassine El MazzouzWe study multivariate Gaussian distributions on local fields such as the field of padic numbers. We introduce the BruhatTits building as a parameter space for Gaussian distributions and study some classic statistical problems in this setting. Finally we study geometric and probabilistic structures of the tropicalization of such distributions.

5:00PM: Symmetry adapted Gram spectrahedronIsabelle ShankarSum of squares (SOS) relaxations are often used to certify nonnegativity of polynomials and are equivalent to solving a semidefinite program (SDP). The feasible region of the SDP for a given polynomial is the Gram spectrahedron. For symmetric polynomials, there are reductions to the problem size that can be done using tools from representation theory. In joint work with Alex Heaton, we used this machinery to disprove a conjecture about symmetric function inequalities. I will give a brief introduction to the theory used. Moreover, I will describe recent work with Serkan Hosten on understanding the geometric structure of the spectrahedra that arise in the study of symmetric SOS polynomials.
