Gradient Flows of Curvature Energies

What defines the “perfect” shape of a knot? Mathematicians approach this question using knot energies—quantitative measures where lower energy corresponds to more “beautiful” configurations. A key example is O’Hara’s repulsion-based energy, inspired by electrostatic forces in higher-dimensional space. But how can we naturally deform a given knot into its optimal form?

This project explored gradient flows for such energies, modeled as steepest descent in an infinite-dimensional landscape. The resulting equations—quasilinear fractional parabolic PDEs—introduced novel mathematical challenges, blending analysis, geometry, and topology. Beyond theoretical intrigue, these energies have applications in biophysics (e.g., protein and DNA modeling) and connections to Willmore surfaces and fractional PDEs, a vibrant area in contemporary mathematics. Funded by the Austrian Science Fund (FWF), this work bridged abstract theory with interdisciplinary frontiers