Vorträge in der Woche 16.05.2022 bis 22.05.2022

Dienstag, 17.05.2022: Ergodic theory meets partial differential equations: The Halmos-von Neumann theorem and the shape of a drum

J. Glück (Wuppertal)

The aim of this talk is to present and compare two classical – but quite different, at first glance – topics from the realms of ergodic theory and partial differential equations: (i) The Halmos–von Neumann theorem in ergodic theory says that certain dynamical systems are uniquely determined up to isomorphism by their spec- tral properties. (ii) The – still wide open – problem under which conditions one can “hear the shape of a drum” in PDE theory asks: when does the spec- trum of the Laplace operator (say, with Dirichlet boundary conditions) on a bounded domain in Rd determine the domain uniquely up to congruence. We discuss that a connection between these two, apparently rather differ- ent, topics can be found by using the language of positive operators: both the conclusion of the Halmos–von Neumann theorem and the desired conclusion of the “shape of a drum” problem can be rephrased as the existence of a pos- itivity preserving linear operator that intertwines certain dynamical systems. The question thus arises whether insights from the proof of the Halmos–von Neumann theorem can be useful to gain a better understanding of the “shape of a drum” problem.

Mittwoch, 18.05.2022: Towards an atlas of Enriques surfaces

Davide Veniani ( Universität Stuttgart )

One of the milestones of algebraic geometry is the classification of algebraic surfaces obtained by Castelnuovo and Enriques, revived and extended by Kodaira, Mumford and Bombieri. In this classification, Enriques surfaces played a fundamental role as the first example of non-rational surfaces with vanishing arithmetic and geometric genus. The original construction by Enriques involves a 10-dimensional family of sextic surfaces in the projective space which are non-normal along the edges of a tetrahedron. In a joint work with G. Martin (Bonn) and G. Mezzedimi (Hannover), we study particular configurations of curves on Enriques surfaces, called (quasi-)elliptic fibrations. As a consequence of our results, we show that all Enriques surfaces arise from Enriques' original construction, as soon as the characteristic of the ground field is not 2. In this talk, I will recollect the 125-year-old history of Enriques surfaces, explain how our work fits in this story, and provide some insights into future projects.

Donnerstag, 19.05.2022: Coordinates are messy

Dr. Melanie Graf (Universität Tübingen)

In General Relativity, an “isolated system at a given instant of time” is modeled as an asymptotically Euclidean initial data set $(M,g,K)$. Such asymptotically Euclidean initial data sets $(M,g,K)$ are characterized by the existence of asymptotic coordinates in which the Riemannian metric $g$ and second fundamental form $K$ decay to the Euclidean metric $\delta$ and to $0$ suitably fast, respectively. Using harmonic coordinates Bartnik showed that (under suitable integrability/decay conditions on their matter densities) the (ADM-)energy, (ADM-)linear momentum and (ADM-)mass of an asymptotically Euclidean initial data set are well-defined. To study the (ADM-)angular momentum and (BORT-)center of mass, however, one usually assumes the existence of Regge-Teitelboim coordinates on the initial data set $(M,g,K)$ in question, i.e. the existence of asymptotically Euclidean coordinates satisfying additional decay assumptions on the odd part of $g$ and the even part of $K$. We will show that, under certain circumstances, harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge-Teitelboim coordinates. This allows us to easily give examples of (vacuum) asymptotically Euclidean initial data sets which do not possess any Regge-Teitelboim coordinates. This is joint work with Carla Cederbaum and Jan Metzger.

Donnerstag, 19.05.2022: Minimale Modelle torischer Hyperflächen

Prof. Dr. Victor Batyrev (Universität Tübingen)

Wir besprechen mit vielen verschiedenen Beispielen die kombinatorische Konstruktion für minimale und kanonische Modelle torischer Hyperflächen und vergleichen diese neue Konstruktion mit dem traditionellen "Minimal Model Program".

Donnerstag, 19.05.2022: The Asymptotic States of Nonlinear Dispersive Equations with Large Initial Data and General Interactions

Prof. Avy Soffer (Rutgers University)

I will describe a new approach to scattering theory, which allows the analysis of interaction terms which are linear and space-time dependent, and nonlinear terms as well. This is based on deriving (exterior) propagation estimates for such equations, which micro-localize the asymptotic states as time goes to infinity. In particular, the free part of the solution concentrates on the propagation set (x=vt), and the localized leftover is characterized in the phase-space as well. The NLS with radial data in three dimensions is considered, and it is shown that besides the free asymptotic wave, in general, the localized part is smooth, and is localized in the region where |x|^2 is less than t. Furthermore, the localized part has a massive core and possibly a halo which may be a self-similar solution. This work is joint with Baoping Liu. This is then followed by new results on the non-radial case and Klein-Gordon equations (Joint works with Xiaoxu Wu)

Freitag, 20.05.2022: 4. FHST Meeting on Geometry and Analysis

Ksenia Fedosova (Freiburg), Stephen Lynch (Tübingen), Hartmut Weiß (Kiel)