Tuesday, April 10, 2018
04/10/2018 - 4:00pm to 5:00pm
We discuss geometry-based statistical learning techniques for performing model reduction and modeling of certain classes of stochastic high-dimensional dynamical systems. We consider two complementary settings. In the first one, we are given long trajectories of a system, e.g. from molecular dynamics, and we estimate, in a robust fashion, an effective number of degrees of freedom of the system, which may vary in the state space of then system, and a local scale where the dynamics is well-approximated by a reduced dynamics with a small number of degrees of freedom. We then use these ideas to produce an approximation to the generator of the system and obtain, via eigenfunctions of an empirical Fokker-Planck equation (constructed from data), reaction coordinates for the system that capture the large time behavior of the dynamics. We present various examples from molecular dynamics illustrating these ideas. In the second setting we only have access to a (large number of expensive) simulators that can return short paths of the stochastic system, and introduce a statistical learning framework for estimatinglocal approximations to the system, that can be (automatically) pieced together to form a fast global reduced model for the system, called ATLAS. ATLAS is guaranteed to be accurate (in the sense of producing stochastic paths whose distribution is close to that of paths generated by the original system) not only at small time scales, but also at large time scales, under suitable assumptions on the dynamics. We discuss applicationsto homogenization of rough diffusions in low and high dimensions, as well as relatively simple systems with separations of time scales, and deterministic chaotic systems in high-dimensions, that are well-approximated by stochastic diffusion-like equations.
04/10/2018 - 4:15pm to 5:15pm
A significant part of the legacy of Ilya Piatetski-Shapiro revolves around the method of integral representations of automorphic L-functions, a method which, in some disguise, is invoked in most occurrences of L-functions in number theory. The goal of these talks will be to describe some developments on this method since the contributions of Piatetski-Shapiro. More precisely, in the first talk I will explain a conjectural framework, and some results, by whichan affine spherical variety gives rise to automorphic distributions thatgeneralize the doubling method, and other Rankin–Selberg constructions of automorphic L-functions. This turns the theory of automorphic L-functions to a local and global problem of harmonic analysis on certain (possibly singular) spaces. In the second talk, I will discuss a further reduction to the quotient stacks associated to Jacquet's relative trace formula. In this setting, integral representations of L-functions blend with Langlands' functoriality conjecture into a broader "relative functoriality"conjecture. I will describe a new way to compare relative trace formulas, which leads to a resolution of this conjecture in rank one.