Vorträge in der Woche 08.07.2024 bis 14.07.2024

Mittwoch, 10.07.2024: Finite element analysis for bulk-surface partial differential equations in evolving domains

Dominik Edelmann

We present results of my dissertation with the title Finite element analysis for bulk-surface partial differential equations in evolving domains. The final outcome is a convergence proof of the spatial semi-discretization of a sophisticated system of partial differential equations describing a model of tissue growth recently derived in Eyles et al. (2019). The work of this third and last paper of my cumulative dissertation is a joint work with Prof. Dr. Christian Lubich and Prof. Dr. Balázs Kovács. The system of equations can be considered as three subsystems that are strongly coupled through the appearance of the mean curvature in both the boundary values of one of the subsystems whose solution appears in the equations for the mean curvature. After this model is presented, we show results on problems similar to the first two subsystems, which are presented in Diffusion equation on a harmonically evolving domain (E., 2022) and in isoparametric finite element analysis of a generalized Robin boundary value problem on curved domains (E. 2021). Then we present how the results and proof techniques used therein can be adapted to the more sophisticated system described above. The first two subsystems can then be treated with these, and the proof of the third subsystem is based on results and techniques of recent works by the above-mentioned collaborators. A sketch of the conference proof is explained and how the proof is clearly separated in a stability part and a consistency part. These have the feature that the stability analysis purely works with matrix-vector equations that arise from discretizing the system and for which no geometric arguments are needed. The consistency proof is then obtained by rewriting the system in a bilinear form setting, and the required estimates use geometric estimates developed in related literature in the past decade. We close with a numerical experiment that illustrates the theoretical results and the dependency of the model on some parameters.

Mittwoch, 10.07.2024: Classifying loops of symmetry-protected states

Prof. Sven Bachmann (University of British Columbia, Canada)

The classification of states of quantum lattice systems is a well-defined mathematical endeavour which started with the discovery of the quantum Hall effect. In this talk, I will discuss the topology of a simple class, the so-called invertible states, which I will define. It is by definition a connected set, and we shall explore its further topological properties. Specifically, I will be interested in what can be identified with its fundamental group; Physically, this is about classifying cycles of physical processes, or pumps. I will present a classification of such loops of invertible state that have a local symmetry, which can be proved to be complete. This is joint work with Wojciech De Roeck, Martin Fraas and Tijl Jappens.

Donnerstag, 11.07.2024: Sobolev conformal structures on closed 3-manifolds

Andoni Royo Abrego (Universität Tübingen)

It is well-known in differential geometry that harmonic coordinates can be used to find the most regular expression for the components of the metric tensor. In fact, work of Sabitov-Shefel and De Turck-Kazdan in the late seventies showed that the the optimal regularity of a Riemannian metric is governed by that of the Ricci tensor. In this talk we will discuss the conformal analogue problem. More precisely, we will study the following question: given a Riemannian metric of limited regularity, does it exist a more regular (even smooth) representative in its conformal class? This question is naturally linked to the Yamabe problem and finds applications in General Relativity.

Donnerstag, 11.07.2024: Towards birational classification of 3-folds determined by empty lattice 4-simplices

Prof. Dr. Victor Batyrev (Universität Tübingen)

Freitag, 12.07.2024: Minimal Lagrangesche Flächen höheren Geschlechts in CP^2

Prof. Dr. Franz Pedit (UMass Amherst)

Jede holomorphe Kurve ist eine Minimalfläche in CP^2, aber sie ist nicht Lagrangesch. Haskins und Kapouleas konstruierten die ersten (und bisher einzigen) Beispiele kompakter minimaler Lagrangescher Flächen in CP^2 ungeraden Geschlechts mittels nichtlinearer PDE Analysis und Klebemethoden. Die Integrabilitätsbedingung für die Existenz minimaler Lagrangescher Flächen ist eine Reduktion der elliptischen Todafeld Gleichung der Liegruppe SU_3, die Tzitzeica Gleichung. Dieser Vortrag erklärt zuerst, wie man diese Gleichung als eine Yang--Mills--Higgs Gleichung auf einem unitären komplexen Rang 3 Bündel über einer kompakten Riemannschen Fläche verstehen kann. Dorfmeister--Pedit---Wu haben gezeigt, wie man solche Gleichungen aus meromorphen Loop Gruppen Zusammenhängen konstruieren kann, falls man die Holonomie Darstellungen dieser Zusammenhänge genügend versteht. Symmetrie Annahmen reduziert das Holonomie Problem auf unitarisierbare Fuchssche Loop SL_3(C) Darstellungen der dreifach punktierten Sphäre. Das Unitarisierungs Problem wird vermöge eines elementaren impliziten Funktionsatzes bewiesen. Das Resultat sind neue Beispiele minimal Lagrangescher Flächen vom Geschlecht (als auch der abstrakten Riemannschen Flaechen Struktur) der Fermatkurven in CP^2.