Vorträge in der Woche 04.12.2017 bis 10.12.2017
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Montag, 04.12.2017: The Stability of Black Holes
Dr. Gustav Holzegel (Imperial College London)
Vorkolloquium um 16 Uhr (moderiert von Jason Ledwidge): I will recall some elementary properties of the linear wave equation on flat and on black hole spacetimes. <br><br>In October, the Nobel Prize in Physics was awarded for the detection of gravitational waves. These waves have been produced by the merger of two black holes, 1.3 billion light years away. <br><br>While we are very far from a mathematical understanding of the interaction of two black holes within Einstein's theory of general relativity, significant progress has been made understanding the dynamics of one black hole near equilibrium (stability). Already here very difficult mathematical challenges arise both from the non-linear nature of the Einstein equations and from the interplay of the intricate black hole geometries with the PDE analysis. <br><br>I will first describe the mathematical notion of a black hole and then report on progress that has been made in the last decade on the analysis of black hole spacetimes. This will include describing some of the techniques from geometry and the theory of hyperbolic partial differential equations that have been developed in the context of the stability problem.
| Uhrzeit: | 17:15 |
| Ort: | N 14 |
| Gruppe: | Kolloquium |
| Einladender: | Cederbaum, Huisken |
Donnerstag, 07.12.2017: On the Stability of the Positive Mass Theorem for Asymptotically Hyperbolic Manifolds
Dr. Armando Cabrera
The Positive Mass theorem asserts that the ADM mass of an asymptotically flat Riemannian manifold with non-negative scalar curvature is non-negative; moreover, the ADM mass equals zero if and only if the manifold is isometric to the Euclidean space. It is then natural to ask about the stability of the Positive Mass theorem, i.e, if the ADM mass is close to zero, is the manifold close to being the Euclidean space? Huang and Lee proved the stability (in the sense of currents) of the Positive Mass theorem for asymptotically flat graphs. We will describe how to use results of Dahl, Gicquaud and Sakovich to adapt Huang and Lee's ideas to the stability of the Positive Mass theorem of asymptotically hyperbolic graphs.
| Uhrzeit: | 14:15 |
| Ort: | S9 |
| Gruppe: | Oberseminar Geometrische Analysis, Differentialgeometrie und Relativitätstheorie |
| Einladender: | Cederbaum, Huisken |
Donnerstag, 07.12.2017: Two numerical schemes for optimal transportation
Soeren Bartels (Universität Freiburg)
Optimal transportation problems define high-dimensional linear programs which are difficult to treat numerically. In the case of a superlinear cost function we use the fact that transport plans are sparse in the sense that they are supported on graphs of convex functions and incorporate optimality conditions to devise an efficient multilevel active set strategy that iteratively determines the discrete support. If transportation cost is proportional to distance we employ a reformulation as a nonlinear partial differential equation. This leads to the Monge-Kantorovich problem which is a constrained minimization problem on Lipschitz continuous functions. We discuss the iterative solution via splitting methods and propose an adaptive mesh refinement strategy based on an a posteriori error estimate for the primal-dual gap. This is joint work with Stephan Hertzog and Patrick Schoen (U Freiburg).
| Uhrzeit: | 14:15 |
| Ort: | N08 |
| Gruppe: | Oberseminar Numerik |
| Einladender: | Lubich, Prohl |
Donnerstag, 07.12.2017: Künnemann’s models of abelian varieties over rank one valuation rings
Dr. Alejandro Soto (Universität Tübingen)
After its discovery in the nineteenth century, abelian varieties have been very important objects in algebraic geometry. In the 70’s and 80’ s Mumford, Faltings-Chai and Raynaud-Bosch-Lütkebohmert developed the theory of degeneration of abelian varieties. It has many applications to arithmetic and geometry. Using this, Künnemann constructed projective regular models of abelian varieties over discrete valuation rings. We will discuss a generalization of these models over arbitrary rank one valuation rings. For this, we will discuss toric geometry over valuation rings, Bertin’s definition of regularity for non-noetherian local rings and the notion of rigid-noetherianess introduced by Fujiwara-Gabber-Kato.
| Uhrzeit: | 14:15 |
| Ort: | N 15 |
| Gruppe: | Oberseminar Algebraische Geometrie |
| Einladender: | Prof. H. Markwig |
Donnerstag, 07.12.2017: Untere Schranken positiver Halbgruppen auf geordneten Banachräumen
M. Wolff (Tübingen)
In der Theorie von Markoff’schen Halbgruppen führte A. Lasota den Begriff der unteren Schranke ein und bewies Stabiltätssätze unter der Voraussetzung der Existenz einer solchen Schranke. Wir verallgemeinern dieses und andere Resultate auf Banachräume mit erzeugendem positiven Kegel und erhalten so die entsprechenden Sätze zum Beispiel für Markoff-Operatoren von Neumann-Algebren. Wir charakterisieren ferner vollständig die Klasse der geordneten Banachräume, auf denen solch eine Theorie entwickelt werden kann.
| Uhrzeit: | 14:15 |
| Ort: | S10 |
| Gruppe: | Oberseminar Funktionalanaysis |
| Einladender: | R. Nagel |
Donnerstag, 07.12.2017: Stability and error analysis for a second-order fast approximation of the 1D Schrödinger Equation under absorbing boundary conditions
Buyang Li (Hong Kong Polytechnic University)
A second-order Crank-Nicolson finite difference method, integrating a fast approximation of the discrete absorbing boundary conditions derived through convolution quadrature, is proposed for solving the one-dimensional Schrödinger equation under absorbing boundary conditions. The fast approximation is based on dyadic decomposition and Gaussian quadrature approximation of the convolution coefficients in the discrete absorbing boundary conditions. It approximates the time convolution in the discrete absorbing boundary conditions by a system of $\mathcal{O}(\log^2(N))$ ordinary differential equations at each time step, where $N$ denotes the total number of time steps. Stability and error estimate are presented for the numerical solutions given by the proposed fast algorithm.
| Uhrzeit: | 15:15 |
| Ort: | N08 |
| Gruppe: | Oberseminar Numerik |
| Einladender: | Lubich, Prohl |
Donnerstag, 07.12.2017: Bulk-edge correspondence for Floquet topological insulators
Dr. Clément Tauber (ETH Zürich)
Floquet topological insulators describe independent electrons on a lattice driven out of equilibrium by a time-periodic Hamiltonian, beyond the usual adiabatic approximation. In dimension two such systems are characterized by integer-valued topological indices associated to the unitary propagator, alternatively in the bulk or at the edge of a sample. In this talk I will give new definitions of the two indices, relying neither on translation invariance nor on averaging, and show that they are equal. In particular weak disorder and defects are intrinsically taken into account. Finally indices can be defined when two driven sample are placed next to one another either in space or in time, and then shown to be equal. The edge index is interpreted as a quantized pumping occurring at the interface with an effective vacuum.
| Uhrzeit: | 16:15 |
| Ort: | N14 |
| Gruppe: | OS MaPhy |
| Einladender: | Hainzl, Keppeler, Porta, Teufel, Tumulka |