Vorträge in der Woche 21.11.2022 bis 27.11.2022

Mittwoch, 23.11.2022: The tropical symplectic Grassmannian

George Balla (Aachen)

For a finite dimensional vector space V, the Grassmannian is the space of all linear subspaces of V of fixed dimension. In 2004, Speyer and Sturmfels introduced the tropicalization of the Grassmannian, and described its polyhedral structure. Since its introduction, several connections to other fields have been made, for example, connection to matroid theory, to toric geometry and representation theory, among others. In this talk, I will discuss a first step towards extending the above picture to the symplectic setting. Given a 2n-dimensional vector space W with a symplectic form w, its linear subspace L is called isotropic if any two vectors in L are orthogonal with respect to w. The symplectic Grassmannian is the space of all isotropic linear subspaces of W of fixed dimension. I will formulate tropical analogues of equivalent characterizations of this space and show that they are not (tropically) equivalent. We will see all implications between them as well as some counter examples. This is joint work with Jorge Alberto Olarte.

Donnerstag, 24.11.2022: Asymptotic-preserving dynamical low-rank approximation for radiation therapy

Jonas Kusch (Universität Innsbruck)

Numerical methods for proton radiation therapy simulate protons moving through and colliding with patient tissue during treatment. Two main solution characteristics need to be resolved in numerical simulations, namely 1) highly peaked radiation waves and 2) diffusion effects that appear due to stiff collision terms at small energies. The need to resolve highly peaked waves as well as the high dimensionality of the phase space can lead to prohibitive memory requirements and computational costs, which limits the applicability of accurate models in clinical use. In this talk we tackle this issue through dynamical low-rank approximation (DLRA) [1]. We present an energy stable DLRA method which is asymptotic–preserving, i.e., it correctly captures diffusion effects when collisions dominate the dynamics. Compared to previous work [2], the use of the “unconventional” basis-update and Galerkin step integrator [3] allows us to prove energy stability of the proposed scheme. Moreover, the corresponding time step restriction captures the hyperbolic and parabolic behavior in different energy regimes. Numerical results demonstrate that the proposed method outperforms conventional methods for radiation treatment planning and yields computational costs that are feasible for clinical use. The presented results have been developed in cooperation with Lukas Einkemmer, Jingwei Hu, Pia Stammer and Niklas Wahl. References [1] O. Koch, C. Lubich. Dynamical low-rank approximation, SIAM Journal on Matrix Analysis and Applications (2007). [2] L. Einkemmer, J. Hu, Y. Wang. An asymptotic-preserving dynamical low-rank method for the multi-scale multi-dimensional linear transport equation, Journal of Computational Physics (2021). [3] G. Geruti, C. Lubich. An unconventional robust integrator for dynamical low-rank approximation, BIT Numerical Mathematics (2022).

Donnerstag, 24.11.2022: Some inequalities involving the ADM mass via linear and nonlinear potential theory

Francesca Oronzio (Università degli Studi di Napoli Federico II)

In this talk, we describe some monotonicity formulas holding along the level sets of suitable p– harmonic functions in asymptotically flat 3–manifolds with a single end, either with or without boundary, having nonnegative scalar curvature. Using such the formulas, we obtain a simple proof of the positive mass theorem and the Riemannian Penrose inequality. The results discussed are obtained by a collaboration with Virginia Agostiniani, Carlo Mantegazza and Lorenzo Mazzieri.

Donnerstag, 24.11.2022: Agmon Estimates on Graphs

Prof. Dr. Matthias Keller (Uni Potsdam)

The philosophy of Agmon says that generalized eigenfunctions "which do not grow too fast in fact decay rapidly". Our setting are Schrödinger operators on graphs and the spectral regime is below the essential spectrum of the operator. The analysis is based on recent work on Hardy and Rellich inequalities on graphs. (This is joint work with Felix Pogorzelski.)

Freitag, 25.11.2022: Differentialgeometrische Strukturen im Kontext der Jacobischen Trennung der Variablen für Hamiltonsche Systeme und Beziehungen zur Riemannschen Theorie der Schockwellen, Teil 2

Prof. Dr. Christoph Bohle (Universität Tübingen)