Veranstaltungsübersicht -- Wintersemester 2024/2025

On a Complete Riemannian Metric on the Space of Closed Embedded Curves joint work with Elias Döhrer and Henrik Schumacher (Chemnitz University of Technology / Univ. of Georgia)

In pursuit of choosing optimal paths in the manifold of closed embedded space curves we introduce a Riemannian metric which is inspired by a self-contact avoiding functional, namely the tangent-point potential. The latter blows up if an embedding degenerates which yields infinite barriers between different isotopy classes.

For finite-dimensional Riemannian manifolds the Hopf—Rinow theorem states that the Heine—Borel property (bounded sets are relatively compact), geodesic completeness (long-time existence of geodesic shooting), and metric completeness of the geodesic distance are equivalent. Moreover, it states that existence of length-minimizing geodesics follows from each of these statements. Albeit the Hopf—Rinow theorem does not hold true in this generality for infinite-dimensional Riemannian manifolds, we can prove all its four assertions for a suitably chosen Riemannian metric on the space of closed embedded curves.

The justification of kinetic equations for long times is a longstanding mathematical challenge; in fact, Hilbert's 6th problem refers specifically to the Boltzmann equation. In my talk I will discuss the case of hard spheres and the very recent progress by Deng & Hani. Finally, I will present results on finite size corrections.

Wir werden ein Modell zur Beschreibung einer Artificial Venus Flytrap formulieren, bei dem das Material als rigid angenommen wird. Während dieses rigide Modell numerisch das Phänomen der Curvature Inversion erfasst, werden wir sehen, dass die Annahme der Rigidity dazu führt, dass die planare Lösung die einzige exakte Lösung ist. Darauf aufbauend werden wir nicht-planare Lösungen betrachten, sobald wir die Rigidity-Annahme fallenlassen. Schließlich werden wir besprechen, wie sich eine Beschreibung einer Limit-Theorie angehen lässt.

Inelastic interaction of granular matter is a common phenomenon in natural processes. A mathematical description of such behaviour is given by a modification of the Boltzmann equation, where the dissipation of kinetic energy during collisions characterises the inelasticity at the particle level.

In this talk, we consider the occurrence of self-similar behaviour in the long-time limit for the one-dimensional inelastic Boltzmann equation. More precisely, we prove that self-similar profiles are unique in the regime of moderately hard potentials. The proof relies on a perturbation argument from the Maxwell model, together with a spectral gap for the corresponding linearised operator.

We will prove the existence of several distinct surfaces of the same given genus that are critical points of the tangent-point energy. The first step of this proof is to pull the tangent-point energy into our comfort zone. The key idea of this step is to describe the surfaces by embeddings of a $2$D manifold $M$ into $\mathbb{R}^3.$ We will define the tangent-point energy on the set of $W^{s,q}$-embeddings, which is an open subset of the Banach space $W^{s,q}(M,\mathbb{R}^3).$ We will discuss this space and characterize the energy space in terms of this regularity. We will see that the tangent-point energy of each $W^{s,q}$-embedding is finite, and each surface with finite energy can be described by a $W^{s,q}$-embedding. Furthermore, we will show that the tangent-point energy is continuously Fréchet differentiable on this domain. Once we have reached this comfortable situation, we will study the energy landscape. By an application of Palais' principle of symmetric criticality and a symmetry argument, we will establish the claimed result.

Two models for the numerical approximation of the elastic movement of inextensible curves are investigated. The configuration with the least possible elastic energy can be approached by employing a gradient flow of the corresponding energy functional. This gradient flow is then discretized using cubic spline functions. First, we examine a scheme that is based on functions that are once globally differentiable, and then we try to recreate that scheme using the twice globally differentiable B-splines. We show the convergence of the discretizations to the continuous problem and compare the performance of the two discretization schemes.