Vorträge in der Woche 23.05.2016 bis 29.05.2016

Dienstag, 24.05.2016: The moonshine module

Julien Sessler

Mittwoch, 25.05.2016: Partial Differential Equations in Geometry

Prof. Dr. Simon Brendle (Stanford University und Columbia University)

A central theme in geometry is the study of manifolds and their curvature. In this lecture series, we will discuss how techniques involving partial differential equations have shed light on several longstanding problems in global differential geometry. In particular, we will focus on the geometry of hypersurfaces, and discuss the isoperimetric inequalities, Alexandrov’s theorem on embedded surfaces in R n of constant mean curvature, as well as our proof of Lawson’s 1969 conjecture concerning embedded minimal tori in S 3 . Time permitting, I will discuss some recent results on the classification of self-similar solutions to geometric flows.

Freitag, 27.05.2016: On the Einstein constraints and the Yamabe problem exterior to a ball

Dr. Stephen McCormick (University of New England)

In this talk we will discuss two separate problems motivated by the Bartnik quasilocal mass. In 2005 Bartnik gave a phase space for the Einstein equations and proved that set of solutions to the constraint equations on an asymptotically flat manifold (without boundary) has a Hilbert manifold structure. Using this framework he demonstrated that critical points of the ADM mass over this Hilbert manifold correspond exactly to stationary initial data. Here we consider the case where an interior boundary is present and discuss the connection to the Bartnik mass. We also discuss the scalar-flat Yamabe problem on an asymptotically flat manifold with boundary, which corresponds to finding time-symmetric vacuum initial data with prescribed boundary conditions.

Freitag, 27.05.2016: Asymptotically flat extensions with non-negative scalar curvature of positive scalar metrics on the 3-sphere

Dr. Armando Cabrera (University of Miami)

Asymptotically flat Riemannian manifolds with non-negative scalar curvature arise naturally as initial data sets for the time-symmetric Cauchy problem for the Einstein equations in general relativity. In this talk, we will describe a geometric procedure to construct an asymptotically flat manifold with non-negative scalar curvature whose ADM mass is arbitrarily close to the optimal value determined by the Riemannian Penrose inequality, while the geometry of its horizon is "far away" from being rotationally symmetric. Specifically, our main result asserts that any metric g of positive scalar curvature on the 3-sphere can be realized as the induced metric on the outermost apparent horizon of a 4-dimensional asymptotically flat manifold with non-negative scalar curvature, such that its ADM mass can be arranged to be arbitrarily close to the optimal value. If time permits, we will discuss some higher dimensional analogous constructions. This talk is based on joint work with Pengzi Miao.