DMS Graduate Student Seminar

Speaker: Dr. James Scott (Auburn University)

Title: New Horizons in Nonlocal Modeling

Abstract: In recent years, continuum models that incorporate nonlocal effects have seen greatly increased use. They have been applied in many areas, including diffusion modeling, image processing, and mechanics. These models are characterized by partial integro-differential equations; that is, in place of partial derivatives, integral operators that act on difference quotients of multi-variable functions are used. In this talk, we will introduce some of these nonlocal models and discuss recent contributions to their mathematically rigorous underpinning across several different contexts. Such contributions include the well-posedness and regularity of nonlocal equations, the robust nature of their discretizations, rigorous characterizations of long-range and other phenomena captured by the equations, and the consistency of nonlocal models with classical models in suitable asymptotic regimes. The contexts include continuum mechanics, semi-supervised learning, fractional PDEs, and coupled local-nonlocal equations. We will conclude with a presentation of some unanswered questions.