Vorträge in der Woche 09.01.2023 bis 15.01.2023

Dienstag, 10.01.2023: 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 spectral 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 spectrum 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 different, 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 positivity 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”.

Mittwoch, 11.01.2023: Gradient Flow Finite Element Discretizations with Energy-Based Adaptivity for the Gross-Pitaevskii Equation

Prof. Benjamin Stamm (Universität Stuttgart)

We present an effective adaptive procedure for the numerical approximation of the steady-state Gross-Pitaevskii equation which consists of a combination of gradient flow iterations and adaptive finite element mesh refinements. The mesh-refinement is solely based on energy minimization. Numerical tests show that this strategy is able to provide highly accurate results, with optimal convergence rates with respect to the number of freedom.

Donnerstag, 12.01.2023: Static vacuum metrics with prescribed Bartnik boundary data

Zhongshan An (University Michigan)

The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as constructing vacuum Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most interesting problems in Riemannian geometry related to general relativity. It is also a problem on the global solvability of a natural geometric system of partial differential equations. In this talk I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.

Donnerstag, 12.01.2023: Diversity & Mathematik. Gibt es das?

Dr. Nicola Oswald (Universität Wuppertal)

In meinem Vortrag werde ich einen praxisorientierten Einstieg zum Forschungsgebiet „Diversity & Mathematik“ geben. Dabei werde ich zunächst auf internationale Forschungsergebnisse zur Fachkultur Mathematik und diversityspezfischen Partizipationsmöglichkeiten in der Mathematik eingehen. Anschließend werde ich vor dem Hintergrund der Situation in Deutschland die Frage diskutieren, ob es hier einen (und welchen) Bedarf an Forschung zu Diversity-Aspekten in der Mathematik gibt. Den Vortrag beschließen möchte ich mit praktischen Beispielen in Bezug auf die Lehre und auf die Lehrenden von Mathematik an Hochschulen.

Freitag, 13.01.2023: Anna Dall’Acqua: On the Willmore flow of tori or revolution; Alex Waldron: Finite-time singularities of 2D harmonic map flow; Marius Müller: Embeddedness-breaking of elastic flows

Prof. Anna Dall'Acqua, Dr. Marius Müller, Prof. Alex Waldron

Anna Dall’Acqua: On the Willmore flow of tori or revolution In this talk we present a stricking relationship between Willmore surfaces of revolution and elastic curves in hyperbolic half-space. Here the term elastic curve refer to a critical point of the energy given by the integral of the curvature squared. In the talk we will discuss this relationship and use it to study long-time existence and asymptotic behavior for the L^2-gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. As in the case of Willmore flow of spheres, we show that if an initial datum has Willmore energy below 8 \pi then the solution of the Willmore flow converges to the Clifford Torus, possibly rescaled and translated. The energy threshold of 8 \pi turns out to be optimal for such a convergence result. The lecture is based on joint work with M. Müller (Univ. Leipzig), R. Schätzle (Univ. Tübingen) and A. Spener (Univ. Ulm). Alex Waldron: Finite-time singularities of 2D harmonic map flow I'll discuss recent work on continuity of the body map at finite-time singularities of 2D harmonic map flow, assuming the initial data is almost-holomorphic (in the energy sense) or the blowup is "strictly type-II." This is relevant to a conjecture of Topping. Time permitting, I'll also discuss uniqueness of subsequential limits at infinite time. Marius Müller: Embeddedness-breaking of elastic flows This talk is based on a joint work with T.Miura (Tokyo) and F. Rupp (Vienna). We study the qualitative behavior of elastic flows of closed curves, i.e. L2-gradient evolutions of the Euler-Bernoulli elastic energy. We are interested in the question of embeddedness-preservation -- Can the evolution of an embedded curve develop self-intersections? In general the answer is 'no', as shown by S. Blatt (2010) for a large class of fourth order geometric flows. We can however expose an (optimal) energy threshold under which evolutions still preserve embeddedness. The optimal threshold has a geometric significance. To understand it we will enter the fantastic world of Euler's elastic curves.

Freitag, 13.01.2023: Konstruktion von CW-Komplexen aus Sullivan-Algebren mit quadratischem Differential

Pascal Zeller (Uni Tübingen)