Abstracts:

Jean Michel Bismut: The hypoelliptic Laplacian and the trace formula

The theory of the hypoelliptic Laplacian produces a family of hypoelliptic operators on the total space X of the tangent bundle of a manifold X, which is supposed to interpolate between the classical Hodge Laplacian (in de Rham or Dolbeault theory), and the generator of the geodesic flow. The hypoelliptic Laplacian is essentially the weighted sum of a harmonic oscillator along the fibre, and of the vector field on X which generates the geodesic flow. When applied to locally symmetric spaces, the deformation is essentially isospectral. We obtain this way a new proof of Selbergs trace formula for cocompact quotients of a symmetric space, and also a precise formula for the evaluation of semisimple orbital integrals. 

L. Saper: Self-dual sheaves and L2-cohomology of locally symmetric spaces
Goresky and MacPherson were motivated to introduce intersection cohomology in part to recover a generalized form of Poincare duality for singular spaces: for each pair of dual perversities, such as the lower middle and the upper middle, there is a nondegenerate pairing between the corresponding intersection cohomology groups. However for singular spaces with even codimension strata, or more generally for Witt spaces, the upper middle and the lower middle theories coincide, yielding a nondegenerate pairing on what is simply called middle perversity intersection cohomology. This enabled Goresky and MacPherson to define an L-class for Witt spaces. For non-Witt spaces, Banagl has shown there exist a well-defined L-class provided there exists a self-dual sheaf that interpolates the lower middle and the upper middle intersection cohomology sheaves. In the case of the reductive Borel-Serre compactification of a Hilbert modular surface, Banagl and Kulkarni show that such a self-dual sheaf exists. In this talk I will address the existence of such self-dual sheaves on the reductive Borel-Serre compactifications of general locally symmetric spaces, a question raised by Banagl and Kulkarni. Note that a completely independent analytic approach to restoring Poincare duality and producing characteristic classes was developed by Cheeger using L2-cohomology. I will also relate the existence of these self-dual sheaves to L2-cohomology.

R. Melrose: Fibrations, corners and pseudodifferential operators
In the category of manifolds with corners there is a natural generalization of the notion of a fibration to that of a b-fibration, which admits controlled degeneracy as the base point approaches the boundary. I will first describe ongoing work with Chris Kottke, Umut Varolgunes and Jonathan Zhu in which the notion of fibre product is discussed in this context. One consequence if such a result is the possibility to introduce corresponding fibre pseudodifferential operators and I will discuss of their properties and applications.

Laurent Clozel: Spectral theory and geometry of compact hyperbolic varieties
This is commom work with N. Bergeron. About 20 years ago Burger, Li and Sarnak announced remarkable conjectures about the spectrum of the Lapalacian for the quotients of hyperbolic spaces by congruence subgroups. Thanks to Arthurs recent resuls on the description of automorphic forms, we have proved these conjectures (some fuzz remains at the top of the spectrum.) By generalizing them to the Hodge Laplacians for higher forms, we obtian remarkable "quasi-Lefschetz" properties of the cohomology of these varieties. Finally, I will describe a new result concerning Thurstons b_1 conjecture in dimension 7

Victor Guillemin: Inverse spectral problems for the semi-classical Schroedinger operator
We will discuss some new inverse spectral results (mainly in 2-dimensions) for the semi-classical Schroedinger operator. These are joint with Alex Uribe and Zuoqin Wang

Lizhen Ji: Geometry and analysis of moduli spaces of Riemann surfaces
In this talk, I will discuss several problems and results on the geometry and analysis on the moduli spaces of Riemann surfaces motivated by corresponding results of locally symmetric spaces.

Erez Lapid: Spectral analysis on locally symmetric spaces

Rafe Mazzeo: Analytic torsion on manifolds with edges
I will report on recent work with Boris Vertman to establish the existence and metric-independence of analytic torsion on compact stratified spaces with simple edge singularities. Surprisingly, in certain cases this turns out to be simpler than for manifolds with isolated conic singularities. I will go on to discuss work in progress concerning some topological applications of these results.

Toshikazu Sunada: Quantum walks in view of discrete geometric analysis
The notion of quantum walks is a ``quantum" version of classical random walks, which is, as indicated by its name, closely related to quantum mechanics. Unlike the case of classical walks depending on imaginary random control by an external device such as the flip of a coin or the cast of a dice, randomness of quantum walks is directly linked to the probabilistic nature of physical states in the quantum system concerned. This talk discusses quantum walks in view of ``geometry of unitary operators". Ideas in discrete geometric analysis, developed to solve various problems in analysis on graphs, are effectively employed to give new insight to the theory.

Andras Vasy: Wave propagation on asymptotically De Sitter and Anti-de Sitter spaces
In this talk I describe the asymptotics of solutions of the wave equation on asymptotically De Sitter and Anti-de Sitter spaces. This is part of a larger program to analyze hyperbolic equations on non-product, non-compact manifolds, similarly to how various types of "ends" have been studied for the Laplacian and other elliptic operators on Riemannian manifolds. The AdS setting is particularly interesting from the point of view of propagation phenomena, since for the conformally related incomplete metric, there are null-geodesics which are tangent to the boundary.