Research Seminar Global Analysis
Prof. Dr. M. Lesch, Prof. Dr. W. Müller
Sommersemester 2010
| 13.07.10 | Fabian Meyer (Bielefeld) Spin^c Dirac operator on T^3 |
| 13.04.10 | Marina Prokhorova (Inst. of Math. and Mechanics / Ural Branch of RAS/MPI) Spectral flow for 1st order selfadjoint elliptic operators on compact surface |
| Abstract: The spectral flow of 1-parameter family of selfadjoint elliptic operators is the algebraic number of operator's eigenvalues intersecting 0. Let A be a 1st order selfadjoint elliptic operator on vector bundle E over compact surface X, B be suitable boundary conditions for A, g be a scalar gauge transformation of E. g transforms A to the operator gA with the same symbol and leave B unchanged. The goal of this talk is to compute the spectral flow along the path (A(t), B) where A(t) connects A with gA in the space of operators with the same symbol. This spectral flow does not change at the deformations of g and (A, B) so it defines homomorphism s from G(X) \otimes H^1 (X, Z) to Z where G(X) is Grothendieck group constructed from the triples (E, A, B) taken up to isomorphism and homotopy. I find group G(X) and homomorphism s for arbitrary compact surface X. Another problem is the computation of the spectral flow along the closed path (A(t), B(t)) where symbol of A(t) and boundary conditions are not constant. It defines homomorphism from some group G_1 (X) to Z. I find G_1 (X) and partially compute this homomorphism. | |
| 20.04.10 | Paul Bressler (MPI) Deformation quantization of gerbes |
| Abstract: After reviewing the basic definitions and classification results concerning deformation quantization algebras (the notion of star-product, relationship to Poisson structures, classification of symplectic deformations) I will explain the difference between algebras and algebroids and the extension of the classification results to the case of deformation quantization algebroids due to Gorokhovsky, Nest, Tsygan and myself. | |
| 27.04.10 | Alexander Gorokhovsky (Boulder/MPI) Local index theorem for projective families |
| Abstract: Mathai, Melrose and Singer stated and proved an extension of the family index theorem to the case of the twisted families of elliptic operators. In this talk I will describe a superconnection proof of this theorem (with some conditions relaxed). This is a joint work with M. Benameur. | |
| 04.05.10 | Victor Kalvin (Academy of Finland) Complex scaling on manifolds with asymptotically cylindrical ends. |
| Abstract: We study spectral properties of the Laplacian on functions by the complex scaling method. This method is well-known as a powerful tool in spectral theory of Schr\"{o}dinger operators in $\Bbb R^n$, however it is not yet well-understood in the frame of the analysis on manifolds. We develop an approach to the complex scaling method and study the Laplacian on a new class of manifolds with asymptotically cylindrical ends. Other known methods, being effective under different geometric assumptions, fail in this setting. We show that there is a certain analogy between spectral theory of Schr\"{o}dinger operators in $\Bbb R^n$ and the analysis on manifolds with asymptotically cylindrical ends. | |
| 11.05.10 | Batu Güneysu (Bonn) Some applications of the Feynman-Kac-Ito formula for magnetic Schrödinger operators on manifolds with bounded geometry |
| Abstract: We explain how a path integral representation of the Schrödinger semigroups corresponding to magnetic Schrödinger operators (with locally square integrable potentials) on manifolds can be used to prove certain regularity properties and to derive bounds on the bottom of the spectrum. | |
| 18.05.10 | Anton Deitmar (Tübingen) Holomorphic torsion and closed geodesics |
| Abstract: Holomorphic torsion of a hermitian locally symmetric space is expressed as a special value of a Selberg-type zeta function, which is formed of monodromy data of closed geodesics. The proof consists of a computation of heat kernel traces and a Lefschetz formula for geodesic actions on locally symmetric spaces. | |
| 08.06.10 | Ognyan Kounchev (Bulgarian Academy of Sciences and IZKS, Bonn) Wavelet Analysis and Microlocal Analysis: sparse representations of Wavefrontsets and Fourier Integral Operators |
| 15.06.10 | Clara Aldana (Universidad de los Andes, Bogota) Ricci flow and the determinant of the Laplacian on non-compact surfaces |
| Abstract: On closed manifolds the determinant of Laplacian is a spectral invariant. For open surfaces the spectrum of the Laplace operator is more complicated and the definition of the determinant does not generalize straightaway. I will briefly explain how to define a renormalized determinant of the Laplace operator in this case. On the other hand, I will talk about Ricci flow on open surfaces and explain how it keeps the decay of the metric at infinity. We use the way in which the determinant varies under a conformal transformation of the metric to show that the determinant increases under Ricci flow. This implies that the largest values are attained at the metric of constant curvature inside a fixed conformal class. This is joint work with Pierre Albin and Frederic Rochon. | |
| 22.06.10 | Mauro Spreafico (ICMC, Sao Paulo) The analytic torsion of the cone over an odd dimensional manifold |
| Abstract: We study the analytic torsion of the cone over an orientable odd dimensional compact connected Riemannian manifold $W$. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a further term that is a rational linear combination of local Riemannian invariants of the boundary. We also prove that this last term coincides with the anomaly boundary term appearing in the Cheeger M\"uller theorem [Ann. Math. 109 (1979)] [Adv. Math. 28 (1978)] for a manifold with boundary, according to Br\"uning and Ma [GAFA 16 (2006)], either in the case that $W$ is an odd sphere or has dimension smaller than six. It follows in particular that the Cheeger M\"uller theorem holds for the cone over an odd dimensional sphere (as conjectured in [J. Math. Pure Ap. 93 (2010)]). We also prove Poincar\'e duality for the analytic torsion of a cone. | |
| 06.07.10 | Leonardo Cano (Bonn) Spectral theory of Laplacians on manifolds with corners of codimension 2 |
Informations
• Tuesday, 14:15 in room 008, Endenicher Allee 60
• Talks last for about an hour plus discussion
Past semester programm
• Wintersemester 2009/2010
• Sommersemester 2009
• Wintersemester 2008/2009
• Sommersemester 2008
• Wintersemester 2007/2008
• Sommersemester 2007
• Wintersemester 2006/2007
• Sommersemester 2006
• Wintersemester 2005/2006