Title: Your dreams may come true with MTP2
Abstract: We study probability distributions that are multivariate totally positive of order two (MTP2). Such distributions appear in various applications from ferromagnetism to Brownian motion tree models used in phylogenetics to factor analysis models used in finance. We first describe some of the intriguing properties of such distributions with respect to conditional independence, graphical models, and parameter estimation. We end with an application to covariance matrix estimation for portfolio selection.
Title: Analysis of high-dimensional systems via pathwise techniques
Abstract: A common motif in high dimensional probability and geometry is that the behavior of objects of interest is often dictated by their marginals onto a fixed number of directions. This is manifested in the fact that several classical functional inequalities are dimension-free (hence, have no explicit dependence on the dimension), the extremizers of those inequalities being functions that only depend on a fixed number of variables. Another related example comes from statistical mechanics, where Gibbs measures can often be decomposed into measures which exhibit a "product-like" structure.
In this talk, we present an analytic approach that helps reveal phenomena of this nature. The approach is based on pathwise analysis: We construct stochastic processes, driven by Brownian motion, associated with the high-dimensional object which allow us to make the object more tractable, for example, through differentiation with respect to time.
I will try to explain how this approach works and will briefly discuss several results that stem from it, including functional inequalities of Gaussian space, concentration inequalities in high-dimensional convexity as well results related to decomposition Gibbs measures into pure states.