Collaborative Research: The Structure of Nonlocal Operators and Applications

该项目旨在回答有关问题，以发展一个更好的数学理解，并建立更复杂的工具，参与非局部相互作用驱动的现象建模。 非局部交互可以通过它与局部交互的对比来说明，在一个简化的图片中，一群代理（比如人）在共享网络中交互。 在局部模型中，人们只能与他们的近邻共享信息，而在非局部模型中，他们可以更广泛地共享信息，可能是在整个网络中，并且可以立即共享信息。 这种简化的图景实际上在许多情况下都表现出来，在这些情况下，非局部范式比局部范式更有成效，而非局部相互作用的引入往往从根本上改变了描述信息传播等方面的基础数学。 该项目旨在通过引入新的数学工具来改进对这种现象的研究。 沿着，该项目将支持指导研究生和本科生通过参与这些主题的研究。它还将创建新的研究生课程材料，为处理数据科学中某些当前主题的非本地工具提供更统一的方法。 在围绕非局部问题的数学领域中，在过去的二十年里出现了一股热潮。 专家们发现他们要解决的问题大致分为两类：（1）为了非定域性本身，应该研究非定域方程的哪些性质;（2）为了将非定域方程与其他领域结合起来，应该研究非定域方程的哪些性质？ 在这个项目中，主要研究人员关注第二类问题。 他们的目标是开发非局部工具，可以应用于一些经典的问题，乍一看，不一定非局部，包括椭圆方程的振荡边界层现象和一个或两个阶段的自由边界问题的问题。关键是，在非定域世界中有一些强大的工具，可以为这些方程提供新的见解或产生新的结果，这些方程以前没有从这个角度进行过研究。这些工具包括但不限于非局部方程的快速增长的正则性理论和非局部方程的弱解理论（仍处于起步阶段）。 通过具有全局比较原理的一般（非线性）算子的表示技术，主要研究人员希望弥合这些先前不相交的方程类的某些方面之间的差距，并使用这种“非局部”桥梁作为新发现的途径。

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.