CAREER: Fractional Partial Differential Equations, Harmonic Analysis, and Their Applications in the Geometric Calculus of Variations and Quantitative Topology

这个项目研究所谓的非局部微分方程结合几何和拓扑效应。涉及非局部微分方程的模型在自然科学中发挥作用，例如物理学（例如，湍流形成）以及有机化学和生物学（例如，DNA和蛋白质打结）。近年来，完善的技术已经发展到治疗非局部微分方面，它的目的是进一步发展和实施这些技术的几何拓扑方面的调查。该项目旨在了解的基本问题是在这种非局部方程下的最佳形状，以及通过非局部模型的几何和拓扑的可控性。该项目为本科生和研究生提供了研究机会。一个特别的方法是与本科生一起开发本项目正在研究的深度几何和拓扑效应的虚拟现实可视化，用于培训和公共宣传。该项目旨在进一步发展分数Sobolev空间和分数变分问题的分析，特别是具有几何背景的问题。该方法，特别是从谐波分析，提供了不同领域的分析，几何和拓扑之间的桥梁，其根本目标是找到并进一步研究这些链接。这个项目的很大一部分是关于分数调和分析在几何偏微分方程和几何变分法中的应用。问题包括非局部自排斥结和曲率能量（存在性和规律性），从物理（适定性）的半波映射方程，和分析的自由边界的变分问题的几何约束（规律性）。第二部分是关于调和分析在定量拓扑中的应用，用于具有小于一个导数的映射（即，Hölder映射，分数Sobolev空间中的映射）：拓扑度的精确估计和海森堡群的拓扑分析研究。在这个项目中集成的是虚拟现实程序的使用和开发，作为一种方法，让本科生参与当前的分析和几何研究课题，帮助研究生发展几何和拓扑概念的直觉，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准。

This project investigates so-called nonlocal differential equations in combination with geometric and topological effects. Models involving nonlocal differential equations play a role in natural sciences such as physics (e.g., turbulence formation) and organic chemistry and biology (e.g., DNA and protein knotting). In recent years, refined techniques have been developed to treat nonlocal differential aspects, and it is the aim of the investigator to further develop and implement these techniques for geometric-topological aspects. The fundamental issues the project aims to understand are optimal shapes under such nonlocal equations, and the controllability of geometry and topology via nonlocal models. Integrated in this project are research opportunities for undergraduate and graduate students. A particular approach is to develop, together with undergraduate students, virtual reality visualizations of the deep geometric and topological effects that this project is investigating, to be used in training and public outreach.The project aims at further developing the analysis of fractional Sobolev spaces and fractional variational problems, especially problems with a geometric background. The methods, in particular from harmonic analysis, provide bridges between different areas of analysis, geometry, and topology, and the underlying goal is to find and further study these links. A large part of this project is concerned with applications of fractional harmonic analysis in geometric partial differential equations and the geometric calculus of variations. Problems include nonlocal self-repulsive knot and curvature energies (existence and regularity), the half-wave map equation motivated from physics (well-posedness), and analysis of the free boundary of variational problems with geometric constraints (regularity). A second part is concerned with application of harmonic analysis in quantitative topology for problems with maps that have less than one derivative (i.e., Hölder maps, maps in fractional Sobolev spaces): sharp estimates of the topological degree and the topological-analytical study of the Heisenberg group are treated. Integrated in this project is the use and development of virtual reality programs as a method for engaging undergraduate students in current research topics in analysis and geometry, for helping graduate students developing intuition for geometrical and topological concepts, and for enabling the broader public to visualize mathematical phenomena.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.