Christian Bär
Sommersemester 2012
Das Seminar behandelt aktuelle Forschungsergebnisse aus der Differentialgeometrie.
Donnerstag 14.15 - 15.45 Uhr
Raum 1.08.0.53
| Datum | Vortrag | Referent | Inhalt |
| 12.04.12 | Rigidity and stability conditions of Einstein manifolds | It is well known that Riemannian metrics of constant curvature are isolated points in the premoduli space of Einstein structures, i.e. the metric cannot be nontrivially deformed such that it stays Einstein. We will see that the same is true if the {tex}$L^{n/2}${/tex}-norm of the Weyl tensor is small enough. In dimensions four and six, the Gauss-Bonnet formula can be used to prove the following: An Einstein metric is rigid if the scalar curvature satisfies an estimate which only involves the Euler characteristic of the manifold. | |
| 19.04.12 | Taylor Expansions and Semiclassical Asymptotics | Matthias Ludewig | In the semiclassical limit, eigenfunctions of Schrödinger operators become "local", i.e. they get concentrated at single points. We develop some calculus of Taylor Expansions on manifolds and use it to solve the appearing differential equations asymptotically. Then we deal with the question in what way these procedures can yield exact results. |
| 26.04.12 | Dirac-harmonic maps, regularization and evolution equations | Dirac-harmonic maps are critical points of an energy functional that involes both the harmonic energy and a spinorial contribution.
We will discuss the evolution equations associated to a regularized version of this energy functional. In the first part of talk we will summarize the results obtained for the case that the domain manifold is a curve or a compact Riemann surface. The second part of the talk will then focus on the bubbling analysis in dimension two and the question if the regularization can be removed. | |
| 31.05.12 | Elektroelastizität | tba | |
| 14.06.12 | An introduction to Seiberg-Witten theory | Ariane Beier | tba |
| 21.06.12 | Some properties of solutions to hypoelliptic equations | Christian Bär | A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 is smooth.
We allow for systems, i.e. the coefficients may be matrices, not necessarily of square size.
This is a huge class of important operators which covers all elliptic, overdetermined elliptic, subelliptic and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence and Riemann's first removable singularity theorem. In the case of constant coefficients we show that Liouville's theorem holds, any bounded solution must be constant and any L^p-solution must vanish. |
| 28.06.12 | String structures | Corbett Redden | String structures are a higher analog of spin structures. In fact, a string structure on a finite-dimensional manifold induces a "spin structure" on the free loop space of the manifold. I will explain this fact, and we will see how they naturally arise when quantizing two-dimensional sigma models. Topologically, a string structure is a trivialization of a degree 4 characteristic class; trivializations in differential cohomology theories (e.g. Cheeger-Simons characters or Deligne cohomology) give useful geometric models for string structures. |
| 12.07.12 | Geometrostatics: the Geometry of Static Spacetimes in General Relativity | Carla Cederbaum | Static spacetimes are geometrically special Lorentzian 4-manifolds possessing a hypersurface-orthogonal timelike Killing vector field. In General Relativity, they are used to model equilibrium configurations of astrophysical objects. We will present several results highlighting special geometric features of static spacetimes as well as the interplay between the geometric, analytic, and physical properties of static spacetimes satisfying the Einstein equations of General Relativity. |