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

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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.