Collaborative Research: The Structure of Nonlocal Operators and Applications

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This project seeks to answer questions about, to develop an improved mathematical understanding of, and to build more sophisticated tools involved in the modeling of phenomena driven by nonlocal interactions. A nonlocal interaction can be illustrated by its contrast to a local interaction in a simplified picture of a population of agents (say, people) interacting across a shared network. A local model is one in which people can share information only with their immediate neighbors, whereas a nonlocal model is one in which they can share information more broadly, possibly across an entire network, and do so instantly. This simplified picture actually manifests itself in many situations in which a nonlocal paradigm is more fruitful than a local one, and the inclusion of nonlocal interactions often fundamentally changes the underlying mathematics that describe aspects such as the propagation of information. This project aims to improve the study of such phenomena through the introduction of new mathematical tools. Along the way, this project will support the mentoring of graduate students and undergraduates through involvement in research on these topics. It will also create new graduate course material that provides a more unified approach to nonlocal tools of use in treating certain current topics in data science. In the realm of mathematics surrounding nonlocal problems, there has been a surge of activity during the past twenty years. Experts have found themselves addressing questions in roughly two categories: (1) What properties of nonlocal equations should be studied for the sake of nonlocality itself; and (2) what properties of nonlocal equations should be studied for the sake of integrating them with other fields? In this project the principal investigators focus on the second type of question. They aim to develop nonlocal tools that can be applied to some classical problems that are not, at first glance, necessarily nonlocal, including questions about oscillatory boundary-layer phenomena for elliptic equations and free-boundary problems of one or two phases. The key point is that there are powerful tools in the nonlocal world that could shed new light upon or produce new results for some of these equations that were not approached from this perspective earlier. Such tools include, but are not limited to, the fast growing regularity theory for nonlocal equations and theory of weak solutions for nonlocal equations (still in its infancy). Through representation techniques for general (nonlinear) operators that enjoy a global comparison principle, the principal investigators hope to bridge the gap between some aspects of these previously disjoint classes of equations and to use this "nonlocal" bridge as a pathway to new discoveries.