Armin Schikorra has recently received a Career Award by the US National Science Foundation (award DMS 2044898) for the proposal “Fractional Partial Differential Equations, Harmonic Analysis, and Their Applications in the Geometric Calculus of Variations and Quantitative Topology” for the period of July 1, 2021 – June 30, 2026. 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.