Gradient Flows of Curvature Energies

Principal Investigator: Simon Blatt

This project has been funded by the FWF (Austrian Science Fund) since summer 2017.

Abstract: What is the most beautiful shape of a knot? What does “beauty” mean in this context? Or more specifically: Is the ball of thread in front of us knotted or not? And: Is there a natural way to transform a very specific knot into this optimal form? To answer these questions, mathematicians have coined the concepts of curvature energies and knot energies over the past two decades and studied these energies. This is an attempt to define a measure for the beauty of a knot. The smaller the energy, the more beautiful the knot. These energies thus provide a pragmatic answer to the first question we asked. Perhaps the best-known energy of this type is based on a very simple physical idea: you want different strands of the knot to be as far apart as possible. The inventor of this energy, Jun O’Hara from Japan, made use of the repulsion effects of electric charge. He distributed a quantum of electrical charge on the knot and calculated the potential energy of this charge distribution. The farther different strands of the knot are, the smaller this potential energy is. However, he had to consider the Coulomb energy in four-dimensional space instead of three-dimensional space in order to actually obtain repulsion effects of different points.this project deals with the last of the above questions: Is there a natural way to transform a very specific knot into its optimal shape? Mathematicians imagine the energy as a mountain landscape and follow the “Direttissima” – the direction of the steepest ascent or, in this case, the steepest descent – like very ambitious mountaineers. The reason for this is certainly that the equations that arise have a new structure – they are so-called quasilinear fractional parabolic differential equations, which are highly non-local and which do not appear to have this structure at first glance. The motivation behind this project is manifold: on the one hand, it is about exciting new mathematics at the edge between such different fields as analysis, geometry and topology. On the other hand, there are interesting cross-connections between some of these energies and the modeling of proteins and DNA, and the modeling of energies is often based on physical ideas. There are connections to other current research topics in mathematics, such as the modeling of membranes, especially Willmore energy, as well as deep topological questions that can perhaps be solved using these techniques. Last but not least, fractional equations are currently one of the trending topics in the mathematics of partial differential equations.