Applications of Equivariant Cohomology

TL;DR

This paper explores the theory and applications of equivariant cohomology, including localization formulas, characteristic class integration, index theory for transversally elliptic operators, and algorithms for computing spline and partition functions.

Contribution

It introduces new localization techniques, applies equivariant cohomology to symplectic quotients and operator indices, and presents algorithms for numerical computation of special functions.

Findings

Localization formulae for equivariant integrals derived

Applications to characteristic class integration on symplectic quotients

Algorithms for computing multivariate spline and vector-partition functions

Abstract

We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic quotients and to indices of transversally elliptic operators. In particular, we state a conjecture for the index of a transversally elliptic operator linked to a Hamiltonian action. In the last part, we describe algorithms for numerical computations of values of multivariate spline functions and of vector-partition functions of classical root systems.