Vorträge in der Woche 02.11.2015 bis 08.11.2015

Dienstag, 03.11.2015: Moore's Ergodicity Theorem

Rupert McCallum

Donnerstag, 05.11.2015: Willmore Tori in Riemannian manifolds

Dr. Andrea Mondino (ETH Zürich)

The Willmore functional of an immersed surface in a Riemannian manifold is the L^2 norm of the mean curvature. The functional appears in many fields of mathematics and science in general: in biology (Helfrich energy), in general relativity (Hawking mass), optics, etc. After an introduction to the topic, in the seminar I will present a joint work with A. Malchiodi and N. Ikoma where we construct Willmore tori in Riemannian manifolds. In particular I will discuss the case the ambient manifold is the Schwarzschild space, one of the most studied manifolds in general relativity.

Donnerstag, 05.11.2015: Über den Beweis des Diskrepanzproblems von Erdös

Uwe Stroinski (Reutlingen)

Um 1932 vermutete Paul Erdös, dass es zu jeder Folge f mit f(n) in {-1,1} und zu jeder natürlichen Zahl c > 0 es natürliche Zahlen k und d gibt, so dass der Absolutbetrag der Reihe mit den allgemeinen Gliedern f(d), f(2d), .... , f(kd) größer als c ist. Diese Vermutung widerstand allen Beweisversuchen bis Terrence Tao im September 2015 einen Beweis vorlegte und damit eine jahrelange, teils sehr intensive Diskussion in mathematischen Blogs zum vorläufigen Abschluss brachte (siehe hierzu etwa http://http://arxiv.org/abs/1509.05363v1 oder http://url9.de/WPt ). Im Vortrag werden Beispiele, Lösungsansätze, erzielte Teilerfolge der letzten Jahre und die schlussendlich erfolgreiche Beweisstrategie von Tao vorgestellt.

Donnerstag, 05.11.2015: Applications of numerical methods to Einstein's and other geometric evolution equations

PD Dr. Oliver Rinne (Albert Einstein Institut Potsdam-Golm)

Numerical methods can provide helpful insights into general relativity well beyond the best-known example of binary black hole / neutron star mergers. In this talk I will give an overview of several projects I have been working on over the past few years. The first part of the talk is concerned with the Einstein equations on very large scales: methods to evolve asymptotically flat spacetimes including conformal infinity. A foliation of spacetime into hypersurfaces of constant mean curvature extending to future null infinity is used, with applications e.g. to late-time power-law tails of matter fields in black hole spacetimes. Conformally flat Cauchy data with a Schwarzschildean end at spatial infinity are constructed numerically, thus realising Corvino's gluing method. The second part of the talk is concerned with the Einstein equations on very small scales: gravitational collapse. I will give examples of how numerical evolutions can be used to probe the weak cosmic censorship conjecture and to study critical phenomena at the threshold of black hole formation. The third part of the talk is concerned with geometric evolution equations other than Einstein's. Examples include a non-local mean curvature type flow used to find static metric extensions in general relativity, and a new method to solve partial differential equations on manifolds with non-trivial spatial topology.