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

本项目研究结合几何和拓扑效应的所谓非局部微分方程解。涉及非局部微分方程的模型在自然科学中发挥着重要作用，如物理学(如湍流形成)、有机化学和生物学(如DNA和蛋白质打结)。近年来，人们发展了精细技术来处理非局部微分方面，进一步发展和实现这些几何拓扑方面的技术是研究者的目标。该项目旨在理解的基本问题是在这种非局部方程下的最优形状，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。一种特别的方法是与本科生一起开发这个项目正在研究的深层几何和拓扑效应的虚拟现实可视化，用于培训和公共出游。该项目旨在进一步发展对分数阶Sobolev空间和分数阶变分问题的分析，特别是具有几何背景的问题。这些方法，特别是来自调和分析的方法，在不同的分析、几何和拓扑领域之间架起了桥梁，基本目标是找到并进一步研究这些联系。这个项目的很大一部分内容涉及分数次调和分析在几何偏微分方程组和几何变分中的应用。问题包括非局部自排斥纽结和曲率能(存在性和正则性)，物理激发的半波映射方程(适定性)，以及几何约束变分问题的自由边界分析(正则性)。第二部分讨论了调和分析在具有少于一阶导数的映射(即Hölder映射、分数次Sobolv空间中的映射)的定量拓扑学中的应用：讨论了拓扑度的精确估计和Heisenberg群的拓扑分析研究。在这个项目中，虚拟现实项目的使用和发展是一种让本科生参与当前分析和几何研究主题的方法，帮助研究生培养几何和拓扑概念的直觉，以及使更广泛的公众能够可视化数学现象。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.