Riemannian nilmanifolds and the trace formula

The trace formula for transversally elliptic operators on Riemannian foliations

The main goal of the paper is to generalize the Duistermaat-Guillemin trace formula to the case of transversally elliptic operators on a compact foliated manifold. First, let us recall briefly the setting of the classical formula. Let P be a positive self-adjoint elliptic pseudodifferential operator of order one on a closed manifold M (for example, P = √ ∆, where ∆ is the Laplace-Beltrami opera...

The Topological Trace Formula

The topological trace formula is a computation of the Lefschetz number of a Hecke correspondence C acting on the weighted cohomology groups, defined in [GHM], of a locally symmetric space X . It expresses this Lefschetz number as a sum of contributions from fixed point components of C on the reductive Borel Serre compactification of X . The proof uses the Lefschetz fixed point formula of [GM2]....

The Kinematic Formula in Riemannian Homogeneous Spaces

Let G be a Lie group and K a compact subgroup of G. Then the homogeneous space G/K has an invariant Riemannian metric and an invariant volume form ΩG. Let M and N be compact submanifolds of G/K, and I(M ∩ gN) an “integral invariant” of the intersection M ∩ gN . Then the integral

Report on the Trace Formula

We report briefly on the present state of the trace formula and some of its applications. This article is a summary of the two hour presentation/discussion on the trace formula. The proposed topic was very broad. It included a recapitulation of the trace formula, past and present, as well as an outlook for its future. The article will treat these matters in only the most concise terms. I includ...

The Iny Ariant Trace Formula