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.