On local boundary conditions for Dirac-type operators
Monday, 3.6.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We give an overview on smooth local boundary conditions for Dirac-type operators, giving existence and non-existence results for local symmetric boundary conditions. We also \n discuss conditions when the boundary conditions are elliptic/regular/Shapiro-Lopatinski (i.e. in particular giving rise to self-adjoint Dirac operators with domain in \(H^1\)). This is joint work with Hanne van den Bosch (Universidad de Chile) and Alejandro Uribe (University of Michigan).
Ricci curvature, metric measure spaces and the Riemannian curvature-dimension condition
Monday, 10.6.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
I explain idea of synthetic Ricci curvature bounds for metric measure spaces and one of their applications in Riemannian geometry.
A necessary condition for zero modes of the Dirac equation
Monday, 17.6.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We will state a necessary condition for the existence of a non-trivial solution of the Dirac equation, which is based on a Euclidean-Sobolev-type inequality. First, we will state the theorem in the flat setting and give an overview of the technical issues of the proof. Afterwards, we will consider and point out the main differences in the not necessarily flat setting. This talk is based on a work by R.Frank and M.Loss.
Fredholmness of the Laplace operator on singular manifolds with pure Neumann Data
Monday, 24.6.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
Given a smooth Riemannian metric \(h\) on \(M\) one can consider the Laplace problem with pure Neumann data: Let \(\bDelta_h\) be the Laplacian given by \(h\) and \(n_h\) be the outer normal of \(\bpartial M\). Exists an \(u\) such that \((\bDelta_hu,\bpartial_{n_h}u)=(F,G)\) for some given data \(F\) and \(G\). There is no well posedness to this problem on singular mandifolds in regular Sobolev spaces but during the talk I will introduce a scale of weighted Sobolev spaces such that it is Fredholm. In the second part of the talk I will give a formula for the Fredholm Index depending on the chosen weight function.