NAMColloquium Geodesically complete Riemannian metrics on the space of knots |  |
Speaker:Dr. Henrik Schumacher, TU Chemnitz
Organizer:Institut für Numerische und Angewandte Mathematik
Details:
Self-avoiding energies were originally constructed to simplify knots and links in $\mathbb{R}^3$. The driving idea was to design barrier functions for the feasible set, the set of curves of prescribed isotopy class. Such functions must blow up whenever a path of curves tries to escape the feasible set, in particular, when self-intersection emerge. This singular behavior makes it challenging to perform numerical optimization for self-avoiding energies, in particular, when ``close'' to the boundary of the feasible set. Motivated by this and inspired by the complete Riemannian metric in the Poincaré model of hyperbolic space, my collaborators and I constructed Riemannian metrics that provide nice preconditioning for the Möbius energy (with Philipp Reiter) and for tangent-point energies (with Keenan Crane). Indeed, these metrics work very well in numerical experiments. In particular, the gradient schemes that we employ require barely any line search in the form of collision detection. That is, with a finite step size we (almost) never bounce into the boundary of the knot space. That led us to the conjecture that these metrics (or some mild modifications) must be geodesically complete. Among other things, this means that the boundary is pushed ``towards infinity''.
In this talk I will present ongoing work with Elias Döhrer and Philipp Reiter. After explaining the notion of geodesic completeness and its applications, I will introduce a certain class of Riemannian metrics on the space of knots. Then I will show that these metrics are closely related to some tangent-point energies and how to exploit this to establish completeness.
In this talk I will present ongoing work with Elias Döhrer and Philipp Reiter. After explaining the notion of geodesic completeness and its applications, I will introduce a certain class of Riemannian metrics on the space of knots. Then I will show that these metrics are closely related to some tangent-point energies and how to exploit this to establish completeness.
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Type:Colloquium
Language:English
Category:Research
Host:Prof. Dr. Max Wardetzky
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Direct link to event:https://events.goettingen-campus.de/event?eventId=12481227
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