Sommersemester 2015
In diesem Seminar werden aktuelle Themen aus der Forschung in der Differentialgeometrie und ihren Nachbargebieten besprochen. Interessenten sind herzlich willkommen.
Donnerstags, 16:15-17:45
Haus 9, Raum 2.06
| Datum | Vortrag | Referent | Inhalt |
| 02.04.15 | Supergeometry in classical field theory | Igor Khavkine | Ordinary (bosonic) classical field theory consists of "field" bundle on a spacetime manifold, a variational PDE on the field sections, its space of solutions (the "phase space", an infinite dimensional manifold), and the algebra of smooth functions ("observables") on the phase space, with an induced Poisson bracket. Fermionic field theory is defined analogously, except that the fibers of the field bundle are allowed to be supermanifolds instead of ordinary manifolds. In the physics literature, fermionic field theories are usually treated in an essentially algebraic way, at the level of the super-Poisson algebra of observables, with its interpretation as the algebra of functions on a phase space supermanifold lost. I will discuss how a modern, functorial formulation of supergeometry allows us to describe the fermionic phase space as a geometric object and to apply tools from analysis and PDE theory to answer some questions about fermionic theories that were difficult to study or even formulate in the algebraic treatment. |
| 23.04.15 | Asymptotic and convergent expansions of heat traces | Michal Eckstein | Asymptotic expansions of heat traces have multifarious applications both in pure mathematics (e.g. index theorems) as well as in mathematical physics (e.g. QFT). Drawing from the theory of general Dirichlet series, I will explain the interplay between heat traces and spectral zeta functions in full generality of operators on a Hilbert space. I will establish general conditions under which an asymptotic expansion of the heat trace exists and discuss its convergence. The general theory will be illustrated with a number of geometrically motivated examples. |
| 30.04.15 | Hyperbolic PDE 1 | Florian Hanisch | We will review the existence and uniqueness results for linear, symmetric hyperbolic systems of PDEs based on energy estimates. We will mostly follow the book by C.D. Sogge, "Lectures on Non-Linear Wave equations" which discusses results on R^{n+1} and only mention more "invariant" results on globally hyperbolic Lorentzian manifolds. If time allows, we will start discussing existence results for quasi-linear systems. |
| 07.05.15 | Hyperbolic PDE 2 | Florian Hanisch | Fortsezung des Vortrags vom 30.04.15 |
| 28.05.15 | An index theorem for Lorentzian manifolds | Christian Bär | We prove an index theorem for the Dirac operator on compact Lorentzian manifolds with spacelike boundary. Unlike in the Riemannian situation, the Dirac operator is not elliptic. But it turns out that under Atiyah-Patodi-Singer boundary conditions, the kernel is finite dimensional and consists of smooth sections. The corresponding index can be expressed by a curvature integral, a boundary transgression integral and the eta-invariant of the boundary operator just as in the Riemannian case. There is a natural physical interpretation in terms of particle-antiparticle creation. This is joint work with Alexander Strohmaier. |
| 04.06.15 | Conformal extendibility and Einstein-Maxwell-Dirac Theory | Olaf Müller | In this talk, we first present the concept of conformal extendibility and its importance in the analysis of the Maxwell-Dirac equations. Then we revise the restrictions conformal extendibility has on the topology and geometry of the standard Cauchy surfaces in the case of standard static spacetimes. Finally, we explain a new approach to present the Einstein-Dirac-Maxwell equations as a variational principle for a function on a Fréchet manifold, and show the existence of a maximal Cauchy development. |
| 11.06.15 | Blockseminar | ||
| 18.06.15 | Flat bundles and K-theory with R/Z coefficients | Sara Azzali | With a flat unitary vector bundle E_a over a closed manifold M one can associate a class a in the K-theory of M with R/Z-coefficients. This class encodes the fact that a flat bundle admits a multiple kE_a which is trivial, and measures the difference between the flat and the trivial connections on kE_a. The construction of the class a is due to by Atiyah, Patodi and Singer, who investigated it in connection with spectral invariants of Dirac operators: they proved that the pairing of a with the K-homology class represented by a Dirac operator gives the value (modulo Z) of the relative eta invariant of D. In this talk, we start by the description of the K-theory groups with real and with R/Z-coefficients, which can be done by means of von Neumann algebras. In fact, the R/Z-K-theory is a relative construction, with respect to the inclusion of the complex numbers into (any) II_1-factor. In this model, we give a canonical construction of the Atiyah-Patodi-Singer class a associated with a flat bundle. Inbetween the ingredients of the construction, Atiyahs L2 index theorem for coverings plays an interesting role, and suggests a generalization of the class a for certain actions of a discrete group on a noncommutative C*-algebra. This is joint work with Paolo Antonini and Georges Skandalis. |
| 25.06.15 | Rigidity results for manifolds with foliated boundary | Georges Habib | In this talk, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow.
Under a suitable curvature assumption depending on the O'Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold.
As a consequence we show that the flow is a local product.
In particular, in the case where solutions of the basic Dirac equation are given by basic Killing spinors, we characterize the geometry of the manifold and the flow. (Joint work with F. El Chami, N. Ginoux, R. Nakad) |
| 02.07.15 | Wellposedness of the Cauchy problem of the linearized Einstein-Klein-Gordon equations | Oliver Lindblad Petersen | We prove existence of a global solution to the linearized Einstein-Klein-Gordon equations, given initial data satisfying the linearized constraint equations. This solution is never unique, as one expects, recalling the non-linear case. The solution is, however, unique up a "linearized diffeomorphism". We thus do not have uniqueness of solution in the sense of linear wave equations and the solution can therefore not depend continuously on initial data in the usual sense. However, we conclude the talk by proving a statement that should be thought of as the continuous dependence of initial data. |
| 23.07.15 | Eta and Zeta functions and Us | Ken Richardson | Eta and zeta functions of geometric operators will be defined, and some elementary properties and relationsships will be described. Applications to classical and more recent work will be presented as time allows. |