Christian Bär
Wintersemester 2014/15
In diesem Seminar werden aktuelle Themen aus der Forschung in der Differentialgeometrie und ihren Nachbargebieten besprochen. Interessenten sind herzlich willkommen.
Donnerstags, 16:15
Haus 8, Raum 0.53
| Datum | Vortrag | Referent | Inhalt |
| 16.10.14 | The heat flow for magnetic geodesics | Florian Hanisch | Magnetic geodesics are curves in a Riemannian manifold which generalize the trajectories of charged particles in magnetic fields to curved geometries. They are characterized by an ODE which is obtained by adding a right hand side to the geodesic equation which represents the force magnetic acting on the curve. We will prove existence of such curves for certain magnetic fields by using a method similar to the heat flow technique for harmonic maps by Eells and Sampson. Building on work bei Koh which garantees long-time existence of the flow, we will establish its (sub)convergence to an actual magnetic geodesic in two cases: 1) the case where the magnetic field admits a global potential and 2) the case where the initial curve of the flow is sufficiently small (and in particular contractible). We will discuss examples illustrating both cases which moreover show that we can not expect convergence of the flows under much more general conditions. This is joint work with Volker Branding. |
| 23.10.14 | Hamiltonian flows and Path Integrals | Matthias Ludewig | Given a Hamiltonian function on the Cotangent bundle of a Riemannian manifold, one associates a Hamiltonian flow to it that describes how classical particles move in phase space. To this data, one can naturally associate a stochastic process that takes into account diffusion, i.e. many interacting particles moving at once. This associates to the Hamiltonian flow on T*M a path integral down on M. We describe how to construct this directly by approximating it with the flow lines of the Hamiltonian flow and show that we constructed the solution operator to the Fokker-Planck equation corresponding to the system. |
| 30.10.14 | Hamiltonian flows and Path Integrals II | Matthias Ludewig | In the previous talk, we described a procedure how to obtain an operator family out of a Hamiltonian flow on the cotangent bundle whose "Chernov-Semigroupisation" (by an Euler type formula) yields the solution operator to the Fokker-Planck equation corresponding to the system, i.e. the equation that describes how particle densities propagate in time. In this talk, we show how this operator family naturally leads to Feynman-type path integrals and the Wiener measure. We use this to recover the Girsanov transformation theorem from the theory of stochastic processes. |
| 13.11.14 | Roe's partitioned manifold theorem with non-trivial fundamental groups | Bernhard Hanke | Positive scalar curvature metrics on non-simply connected closed spin manifolds are obstructed by an index invariant in the K-theory of a certain operator algebra associated to the fundamental group. We generalize Roe's partitioned manifold theorem to non-simply connected manifolds. From this we derive an index obstruction to positive scalar curvature metrics on closed spin manifolds, which lives in the K-theory of an operator algebra associated to the fundamental group of a codimension-2 submanifold. |
| 20.11.14 | The characteristic Cauchy problem for wave equations on manifolds | Christian Bär | The characteristic Cauchy problem for linear wave equations consists of imposing initial values for the solution on a characteristic hypersurface instead of initial values for the function and its normal derivative on a spacelike Cauchy hypersurface. After an introduction to wave equations on Lorentzian manifolds we show that this problem is well posed under suitable assumptions. This is joint work with Roger Tagne Wafo and it extends classical results by Hörmander. |
| 27.11.14 | SJ and FP states and Quantum Inequalities for quantized fields in curved spacetimes | Rainer Verch | Recently, SJ states and FP states have been proposed as "local vacuum states" for quantized scalar fields, and quantized Dirac fields, respectively, on curved spacetimes.
Properties of these states will be discussed in the talk; results obtained so far show that for certain non-geodesically complete spacetimes, SJ states and FP in general fail to fulfill the microlocal spectrum condition and therefore are not suitable as physical states.
Certain variations providing "mollified" versions of SJ and FP states however turn out to fulfill the microlocal spectrum condition. Quantum energy inequalities are lower bounds on locally time-averaged energy quantities of quantum fields; such bounds have beeen establishes for quantizied scalar fields and quantized Dirac fields for the sets of states fulfilling the microlocal spectrum condition. Some of these results will be summarized in the talk. |
| 18.12.11 | Chern-Simons invariants and 5-manifolds | Christian Becker | In this talk, we will give a brief overview over recent results due to Steve Rosenberg and collaborators on diffeomorphism groups of certain 5-manifolds. In a series of papers, Rosenberg together with several co-authors introduced Chern-Simons invariants for certain infinite dimensional groups. In the recent papers, these invariants are used to show that for certain 5-manifolds, the fundamental group of the diffeomorphism group is infinite. |
| 08.01.15 | Spektrum auf flachen Tori und Sphären | Franzi Beitz | Gegeben sei der Operator \alpha d\delta + \beta\delta d für \alpha,\beta> 0 auf Einsformen auf einer kompakten riemannschen Mannigfaltigkeit. Wir berechnen das Spektrum auf dem n-dimensionalen flachen Torus und der Sphäre. |
| 15.01.15 | On (interesting) quotients of noncommutative manifolds | Andrzej Sitarz | There is a plenitude of interesting manifolds that are obtained as quotients by an action of a finite group.
A good example are quotients of tori, which are so-called Bieberbach manifolds or quotients of spheres (lens spaces).
In the classical case there are only two 2-dimensional Bieberbach manifolds, however, it appears that in the noncommutative case a pillow (an orbifold with four corners) is more regular. I will discuss the problem how to distinguish noncommutative "manifolds" from "orbifolds" using various methods of noncommutative geometry. A typical example to illustrate these methods are the pillows and Moyal cones. |
| 22.01.15 | Vacuum static solutions of the Einstein equations (in dim 3) | Martin Reiris | The first uniqueness result for the Schwarzschild solution was established by Israel in 1967. It was later improved by Robinson (1977) and Bunting-Masood-ul Alam (1987) to the extent that now the Schwarzschild solution is known to be unique among the asymptotically flat static solutions with compact (but non necessarily connected horizons). In this talk we will show that the hypothesis of asymptotic flatness can be replaced by a topological condition (that outside a compact set there is a connected component diffeomorphic to R^3 minus a ball). The Korotkin-Nicolai black hole shows that this topological condition cannot be relaxed in any way. This result opens a window to classify all regular static solutions in dimension 3, which could impact in several areas in differential geometry. |
| 29.01.15 | Vanishing of the eta residue density for projective vector bundles | Alexander Strohmaier | I will explain a short argument that implies the vanishing of the eta residue density for odd dimensional manifolds. This argument is algebraic and therefore generalizes the original results by Wodzicki and Gilkey to the projective case. The talk will start with a very elementary introduction to projective bundles, pseudodifferential operators and twisted K-theory. |
| 05.02.15 | Fundamental solutions of the Klein-Gordon operator in Minkowski space | Max Lewandowski | We calculate the 2-point-distribution and the causal propagator of the Klein-Gordon field in Minkowski space, derive fundamental solutions of the Klein Gordon operator and compare them with analogous expressions including Riesz' distributions. |