Colloquium Abstracts Spring 2023

Abstracts will be posted here for the colloquium talks when they are available.

Zhen-Qing Chen

University of Washington
February 9, 2023

Boundary Harnack Principle for Non-local Operators

It is well known that scale invariant boundary Harnack inequality holds for Laplacian \Delta on uniform domains and holds for fractional Laplacians \Delta^s on any open sets. It has been an open problem whether the scale-invariant boundary Harnack inequality holds on bounded Lipschitz domains for Levy processes with Gaussian components such as the independent sum of a Brownian motion and an isotropic stable process (which corresponds to \Delta + \Delta^s).

In this talk, after an introduction of boundary Harnack inequality and some of its history, I will present a necessary and sufficient condition for the scale-invariant boundary Harnack inequality to hold for a class of non-local operators on metric measure spaces. This result will then be applied to give a sufficient geometric condition for the scale-invariant boundary Harnack inequality to hold for subordinate Brownian motion on bounded Lipschitz domains in Euclidean spaces. This condition is almost optimal and a counterexample will be given showing that the scale-invariant BHP may fail on some bounded Lipschitz domains with large Lipschitz constants. Based on joint work with Jie-Ming Wang.

Hailing Liu

Iowa State University
March 10, 2023

Gradient Methods with Energy for Deep Learning Problems

We will present some mathematical problems encountered in deep learning models. The results include optimal control of selection dynamics in deep neural networks, and gradient methods with energy. Some of the computational questions that will be addressed have a more general interest in engineering and sciences.

Kevin Woods

Oberlin College
March 23, 2013

A Plethora of Polynomials: A Toolbox for Counting Problems Using Presburger Arithmetic

A wide variety of problems in combinatorics and discrete optimization depend on counting the set S of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour through numerous problems of this type. In particular, we consider families of such sets S_t depending on one or more integer parameters t, and analyze the behavior of the function f(t)=size of S_t. In the examples that we investigate, this
function exhibits surprising polynomial-like behavior. We describe settings (namely, variations on Presburger Arithmetic) where this polynomial-like behavior must provably hold. The plethora of examples illustrates the framework in which this behavior occurs, helping us create a toolbox for counting problems like these.

Jin Feng

University of Kansas
April 6, 2023

On a Weak Hydrodynamic Limit Theory

To understand mechanical origin of probability models in statistical and continuum mechanics, it is useful to study hydrodynamic limit for interacting particles following deterministic Hamiltonian dynamics. Traditional approach on such a program faces many difficulties. One of them is about  rigorous justification of canonical type ensembles. This is because relevant deterministic ergodic theory is still largely out of reach. Another huge barrier is on making sense of rigorous meaning of hyperbolic conservation laws. Such PDEs are used to express F=ma and thermodynamic relations in the continuum.

We examine a new line of thoughts by formulating the hydrodynamic limit program as a multi-scale abstract Hamilton-Jacobi theory in space of probability measures.

This talk will focus on derivation of an isentropic model. Through mass transport calculus, we develop tools to reduce the hydrodynamic problem to known results on finite dimensional weak KAM (Kolmogorov-Arnold-Moser) theory, showing sufficiency of using a weak version ergodic results on micro-canonical ensembles, instead of the canonical ones. We will also reply on recent progress of viscosity solution theory for abstract Hamilton-Jacobi equation in space of probability measures, using a calculus in Alexandrov metric spaces. Such approach gives a weak and indirect characterization on evolution of the limiting continuum model using generating-function  formalism at the level of canonical transformation in calculus of variations. It avoids the use of hyperbolic systems of PDEs, which operates at the level of abstract Euler-Lagrange equations of action functionals.

All together, these techniques enable us to realize a weaker but rigorous version of the hydrodynamic limit program in some nontrivial cases.

Andrzej Swiech

Georgia Tech University
May 4, 2023

Finite Dimensional Approximations of Hamilton-Jacobi-Bellman Equations in Spaces of Probability Measures and Stochastic Optimal Control of Particle Systems

We will discuss recent results about a class of Hamilton-Jacobi-Bellman (HJB) equations in spaces of probability measures that arise in the study of stochastic optimal control problems for systems of n particles with common noise, interacting through their empirical measures. We will present a procedure to show that the value functions u_n of n particle problems, when converted to functions of the empirical measures, converge as n→∞ uniformly on bounded sets in the Wasserstein space of probability measures to a function V, which is the unique viscosity solution of the limiting HJB equation in the Wasserstein space. The limiting HJB equation is interpreted in its "lifted" form in a Hilbert space, a technique introduced by P.L. Lions. The proofs of the convergence of u_n to V use PDE viscosity solution techniques. An advantage of this approach is that the lifted function U of V is the value function of a stochastic optimal control problem in the Hilbert space. We will discuss how, using Hilbert space and classical stochastic optimal control techniques, one can show that U is regular and there exists an optimal feedback control. We then characterize V as the value function of a stochastic optimal control problem in the Wasserstein space. The talk will also contain an overview of existing works and various approaches to partial differential equations in abstract spaces, including spaces of probability measures and Hilbert spaces.