Christian Bär
Sommersemester 2016
In diesem Seminar werden aktuelle Themen aus der Forschung in der Differentialgeometrie und ihren Nachbargebieten besprochen. Interessenten sind herzlich willkommen.
Donnerstags, 16:15
Campus Golm, Haus 9, Raum 0.14
| Datum | Vortrag | Referent | Inhalt |
| 14.04.16 | Construction of solutions of wave equations on global hyperbolic Lorentzian manifolds via Hadamard expansion | Max Lewandowski | I will start again with solutions of d'Alembert's equation on n-dimensional Minkowski space fulfilling Wightman's Axioms as a prototype with regard to general wave equations on global hyperbolic Lorentzian manifolds. With an appropriate classification of distributions, which are invariant with respect to the special orthochronous Lorentz group, one can decompose these solutions into Riesz' distributions, which are based on a well understood theory, and a linear combination of symmetric and all over supported fundamental solutions of the d'Alembert operator. In order to establish a similar construction as for the Riesz distributions we will embed them into another family of holomorphic distributions, translate this family into geodesically convex subsets of global hyperbolic Lorentzian manifolds and eventually consider Hadamard's expansion in order to obtain symmetric and all over supported fundamental solutions to general normal hyperbolic operators. Thereby the local theory for the construction of solutions of wave equations is established and if there is enough time we will see that the singular part for example of the solution of Klein-Gordon's equation is in fact composed of these ingredients. |
| 21.04.16 | Generalised Hyperbolicity for Singular Spacetimes | Yafet Sanchez Sanchez | A desirable property of any spacetime is that the evolution of any physical field is locally well-defined.
For smooth spacetimes this is guaranteed by standard local well-posedness results.
Moreover, there are physically reasonable spacetimes for which the initial value problem is well-posed but the spacetime has low differentiability. In this talk we will show that in certain spacetimes with hypersurface and stringlike singularities one still has local well-posedness of the wave equation in the Sobolev space $H^1$. This function space is chosen as it allows us to define the energy-momentum tensor of the field distributionally. It is also the one needed for solutions of the linearised Einstein equations. The methods we employ are therefore also relevant to finding low regularity solutions of the Einstein equations which, as shown by Dafermos, is an important issue when considering Strong Cosmic Censorship. Motivated by this work we propose a definition of a strong gravitational singularity as an obstruction to the evolution of a test field rather than the usual definition as an obstruction to the evolution of a test particle along a causal geodesic. This definition has the advantage that it is directly related to the physical effect of the singularity on the field (the energy-momentum tensor fails to be Integrable) and also that it may be applied to situations where the regularity of the metric falls below $C^{1,1}$ where one no longer has existence and uniqueness of geodesics. This is joint work with James Vickers. |
| 28.04.16 | The Cauchy problem of the linearised Einstein equations | Oliver Lindblad Petersen | In this talk, we consider the Cauchy problem of the linearised Einstein equation on smooth globally hyperbolic spacetimes, satisfying the non-linear Einstein equation. Given smooth or distributional initial data (of arbitrary Sobolev regularity) specified on a spacelike Cauchy hypersurface, we show that there is a globally defined solution, which is unique up to gauge. Hence the solutions, modulo gauge, are in one-to-one correspondence with the initial data, modulo gauge producing initial data. If the Cauchy hypersurface is closed, this correspondence is actually an isomorphism of topological vector spaces. Therefore, in this sense, the Cauchy problem of the linearised Einstein equation is well posed. |
| 12.05.16 | Isotropic hyperelastic shells – Geometry, Physics, Examples | Claudia Grabs | We consider two-dimensional shells made of isotropic and hyperelastic material. First, the basic equations of elasticity are recalled, with special emphasis on the constitutive laws. Subsequently we want to model some example configurations. If the displacement of the boundary is prescribed, the problem is to find a static solution to the equations of equilibrium. Parts of the lecture will be underlined by SAGE programs. |
| 19.05.16 | Semilinear wave equations and the Yamabe problem on Lorentzian manifolds | Viktoria Rothe | In this talk we will consider the Cauchy problem for semilinear wave equations on globally hyperbolic Lorentzian manifolds. We will examine under which conditions we obtain time-global solutions for small initial data and time-local solutions for arbitrary initial data. Subsequently we will analyze if these existence results can be applied to the Yamabe Problem on Lorentzian manifolds. |
| 26.05.16 | On the positive mass conjecture for closed Riemannian manifolds | Andreas Hermann | Let (M,g) be a closed Riemannian manifold such that all eigenvalues of the conformal Laplace operator L_g of g are strictly positive and such that g is flat on an open neighborhood of a point p.
The constant term in the expansion of the Green function of L_g at p is called the mass of (M,g) at p.
It is an open conjecture that under the assumptions above the mass is non-negative and that it is zero if and only if (M,g) is conformally diffeomorphic to the round sphere. In this talk we introduce the mass of a more general class of elliptic operators of the form \Delta_g+f, where f is a smooth function on M vanishing on an open neighborhood of p, and we discuss some properties of this mass. This is joint work with Emmanuel Humbert. |
| 09.06.16 | The scalar wave equation with(out) topological charge on Schwarzschild and Reissner-Nordström black hole spacetimes | Ariane Beier | The aim of this talk is to present the results of the analysis of the separated solutions of the scalar wave equation with and without topological charge on Schwarzschild and Reissner-Nordström black hole spacetimes with Sturm-Liouville methods. With respect to these methods, a complete characterisation of the separated solutions of the scalar wave equation (without topological charge) is given on the exterior as well as the interior of the Schwarzschild background. Since the other cases are still work in progress, only the available results and occuring issues are presented. |
| 29.09.16 | Gamma-structures on symmetric spaces | Bernhard Hanke | Gamma-structures are weak forms of multiplications on closed oriented manifolds.
As shown by Hopf the rational cohomology algebras of manifolds admitting Gamma-structures are free over odd degree generators.
We prove that this condition is also sufficient for the existence of Gamma-structures on manifolds which are nilpotent in the sense of homotopy theory.
This includes homogeneous spaces with connected isotropy groups. Passing to a more geometric perspective we show that on compact oriented Riemannian symmetric spaces with connected isotropy groups and free rational cohomology algebras the canonical products given by geodesic symmetries define Gamma-structures. |
| 30.06.16 | Dirac operators on foliations | Ken Richardson | Given a foliation on a closed Riemannian manifold, the transversal Dirac operator is a Dirac operator that differentiates only in the directions normal to the foliation and is thereby transversally elliptic. This operator has nice properties and a well-defined index when the metric is bundle-like --- that is, when the foliation locally has the structure of a Riemannian submersion. We will show some standard techniques used to do analysis on these operators and discuss some known results. The talk will contain some joint work with G. Habib and with F. W. Kamber and J. Brüning. |
| 14.07.16 | Boundary value problems for Dirac operators | Sebastian Hannes | The talk deals with boundary value problems for dirac type operators on a complete Riemannian manifold with compact boundary.
After a short introduction to Dirac operators on Riemanian manifolds, suitable boundary conditions to ensure Fredholm property of the involved operators will be introduced.
Some results concerning index calculation and spectral theory for these operators will be discussed. References: Bär, Ballmann: Guide to boundary value problems for Dirac-type operators, Boundary value problems for elliptic differential operators of first order |