Research Seminar Global Analysis
Prof. Dr. M. Lesch, Prof. Dr. W. Müller
Winter semester 2013
Talks
5.11.2013 | Tobias Finis (FU Berlin) Title: Congruence subgroups of arithmetic lattices and the limit multiplicity property |
Abstract: We study the limiting behavior of the discrete spectra of congruence subgroups of an irreducible arithmetic lattice in a semisimple Lie group $G$. Assuming that the subgroups in question do not contain any non-trivial central elements, one expects their spectra to converge to the Plancherel measure of $G$ (the limit multiplicity property). We are able to prove this property for the lattices ${\rm SL} (n, \mathfrak{o}_F)$, where $F$ is a number field, and obtain conditional results in the general case. The focus lies on the case of non-compact quotients, where the spectra have a continuous part. There are two main parts of the proof, which is based on Arthur's trace formula. First, we prove some general results on congruence subgroups of arithmetic lattices and derive bounds on the number of fixed points of non-central elements on the corresponding finite permutation representations. Second, we reduce the control of the continuous spectrum to two conjectural properties of intertwining operators, one global and one local, which we can verify for the groups ${\rm GL} (n)$ and ${\rm SL} (n)$. This is joint work with Erez Lapid (Rehovot and Jerusalem) and Werner M"uller (Bonn). | |
03.12.2013 | Farzad Fathizadeh (IHES Paris) in the seminar room N0.007 Title: Scalar curvature and Einstein-Hilbert action for noncommutative tori |
Abstract: After the seminal work of A. Connes and P. Tretkoff on the Gauss-Bonnet theorem for the noncommutative two-torus, there have been significant developments in understanding the local differential geometry of these noncommutative spaces equipped with curved metrics. In this talk, I will review a series of joint works with M. Khalkhali, in which we extend this result to general translation invariant conformal structures on noncommutative two-tori, compute the scalar curvature, and prove the analogue of Weyl's law and Connes' trace theorem. Our final formula for the curvature matches precisely with the one computed independently by A. Connes and H. Moscovici. I will then report on our recent work on the computation of scalar curvature for noncommutative four-tori (which involves intricacies due to violation of the Kaehler condition). We show that metrics with constant curvature are extrema of the analogue of the Einstein-Hilbert action. A purely noncommutative feature in these works is the appearance of the modular automorphism from Tomita-Takesaki theory in the computations and final formulas for curvature | |
17.12.2013 | Xiaonan Ma (Paris) Title: Exponential Estimate for the asymptotics of Bergman kernels |
Abstract: We will explain an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. As applications, we will give Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line bundles. This is a joint work with George Marinescu. | |
14.01.2014 | Shu Shen (Orsay) Title: Hypoelliptic Laplacian, Witten deformation and analytic torsion |
Abstract: in this talk, after a short introduction to the hypoelliptic Laplacian in the de Rham theory, the hypoelliptic analytic torsion and the hypoelliptic Ray-Singer metric, we prove a hypoelliptic version of the Cheeger-Müller theorem using a Witten-like deformation. Then we deduce the result due to Bismut and Lebeau which states that the hypoelliptic Ray-Singer metric coincides with the classical elliptic Ray-Singer metric. | |
Informations
• Tuesday, 14:15 in room 008, Endenicher Allee 60
• Talks last for about an hour plus discussion
Past semester programm
• Summer semester 2013
• Winter semester 2012/2013
• Sommersemester 2012
• Wintersemester 2011/2012
• Sommersemester 2011
• Wintersemester 2010/2011
• Sommersemester 2010
• Wintersemester 2009/2010
• Sommersemester 2009
• Wintersemester 2008/2009
• Sommersemester 2008
• Wintersemester 2007/2008
• Sommersemester 2007
• Wintersemester 2006/2007
• Sommersemester 2006
• Wintersemester 2005/2006