Variational problems for knotted curves and surfaces: how repulsive energies come into play

05.06.2024, 14:00 - 16:00  –  Campus Golm, Building 9, Room 2.22 and via Zoom
Institutskolloquium

Heiko von der Mosel (University of Aachen), Henrik Schumacher (University of Georgia)

14:00  Heiko von der Mosel (University of Aachen):  On the Tangent-Point Energy.
14:45  Tea and Coffee Break
15:15  Henrik Schumacher (University of Georgia):  Numerical Optimization of the Tangent-Point Energy.

Heiko von der Mosel (University of Aachen):  On the Tangent-Point Energy.

Abstract:     In geometric knot theory one considers various types of self-avoidance energies to deal with variational problems for knotted curves and submanifolds. In this talk I will focus on the tangent-point energy, describe its self-avoidance mechanism and its regularizing properties. As key results I will highlight geometric Morrey embeddings theorems, existence of minimizers and critical points, and finiteness theorems for isotopy types of curves and surfaces.

Henrik Schumacher (University of Georgia):  Numerical Optimization of the Tangent-Point Energy.

Abstract:     Repulsive energies like the tangent-point energy were originally constructed to simplify knots in space. The driving idea was to design energies that blow up to infinity when a time-dependent family of knots develops a self-intersection. Thus, downward gradient flows should simplify a given knot without escaping its knot class. In principle, the same approach can be taken for optimizing the shape of knotted surfaces. Alas, one faces several computational challenges, the most difficult being: (i) straight-forward discretizations of the energy via Riemann sums have a computational cost that grows quadratically with the number of degrees of freedom, which renders them barely useful for practical purposes; (ii) the discretized energies lead to very ill-conditioned optimization problems, and black box optimization methods perform poorly on them.
     In this talk I will address issues (i) and (ii) after which (and most importantly), I will reap the rewards of these efforts and present a couple of videos that employ gradient flows of the tangent-point energy to visualize some stunning facts from the field of topology.

Wenn Sie digital an den Vorträgen teilnehmen möchten, wenden Sie sich bitte an Christian Molle molle @ uni-potsdam.de, um die Zugangsdaten zu erhalten.