Renato Velozo (University of Cambridge): Stability of Schwarzschild for the spherically symmetric Einstein--massless Vlasov system. Oberseminar Topics in General Relativity
Tuesday, 24.05.2022 12:00 im Raum SRZ204
The Einstein--massless Vlasov system is a relevant model in the study of collisionless many particle systems in general relativity. In this talk, I will present a stability result for the exterior of Schwarzschild as a solution of this system assuming spherical symmetry. We exploit the hyperbolicity of the geodesic flow around the black hole to obtain decay of the stress energy momentum tensor, despite the presence of trapped null geodesics. The main result requires a precise understanding of radial derivatives of the energy momentum tensor, which we estimate using Jacobi fields on the tangent bundle in terms of the Sasaki metric.
Dr. Melanie Graf (Universität Tübingen): Coordinates are messy. Oberseminar Topics in General Relativity
Tuesday, 31.05.2022 12:00 im Raum SRZ204
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 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.
Mittagsseminar zur Arithmetik: Guido Kings (Univ. Regensburg): Eisenstein-Kronecker cohomology classes and critical values of Hecke L-functions
Tuesday, 31.05.2022 14:15 im Raum via Zoom
(joint with Johannes Sprang). Let K be a CM ?eld and L/K be an extension of degree n and ? be an algebraic critical Hecke character of L. Then we show that the L-value L(?, 0) divided by a carefully normalized Shimura-Katz period is integral and construct in the ordinary case a p-adic L-function for ?. This generalizes results by Damerell, Shimura and Katz for CM ?elds (L = K) and settles all remaining open cases of algebraicity of critical Hecke L-values.
Our method relies on the construction of new equivariant Eisenstein-Kronecker cohomology classes using the completion of the Poincar´e bun-dle. Here we were inspired by work of Bannai-Kobayshi in the imaginary quadratic case. Our method is even in the classical CM case completely dif-ferent from the approaches by Shimura and Katz. With a technique, which was pioneered in Sprang?s thesis, this approach leads without any computa-tions directly to the construction of measures giving the p-adic L-functions for ?.
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