On a Complete Riemannian Metric on the Space of Closed Embedded Curves
Tuesday, 29.10.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
On a Complete Riemannian Metric on the Space of Closed Embedded Curves\njoint work with Elias Döhrer and Henrik Schumacher (Chemnitz University\nof Technology / Univ. of Georgia)\n\nIn pursuit of choosing optimal paths in the manifold of closed embedded\nspace curves we introduce a Riemannian metric which is inspired by a\nself-contact avoiding functional, namely the tangent-point potential.\nThe latter blows up if an embedding degenerates which yields infinite\nbarriers between different isotopy classes.\n\nFor finite-dimensional Riemannian manifolds the Hopf—Rinow theorem\nstates that the Heine—Borel property (bounded sets are relatively\ncompact), geodesic completeness (long-time existence of geodesic\nshooting), and metric completeness of the geodesic distance are\nequivalent. Moreover, it states that existence of length-minimizing\ngeodesics follows from each of these statements. Albeit the Hopf—Rinow\ntheorem does not hold true in this generality for infinite-dimensional\nRiemannian manifolds, we can prove all its four assertions for a\nsuitably chosen Riemannian metric on the space of closed embedded\ncurves.\n
Rigorous justification of kinetic equations: Recent progress and finite size corrections
Tuesday, 12.11.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
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.
Das Rigidity Phänomen einer Artificial Venus Flytrap
Tuesday, 26.11.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
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.
The 1d inelastic Boltzmann equation for moderately hard potentials
Tuesday, 17.12.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
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.\n\nIn 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.
The tangent-point energy for surfaces and its symmetric critical points
Tuesday, 28.1.25, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We will prove the existence of several distinct surfaces of the same given genus that are critical points of the tangent-point energy.\nThe 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 \(\bmathbb{R}^3.\) We will define the tangent-point energy on the set of \(W^{s,q}\)-embeddings, which is an open subset of the Banach space \(W^{s,q}(M,\bmathbb{R}^3).\) 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.\nOnce 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.
B-Spline Discretization of Inextensible Curves
Tuesday, 4.2.25, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
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.\n