Applications of Equivariant Cohomology

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.

This text is to be submitted to the Proceedings of the International Congress of Mathematicians, Madrid, August 2006. Minor changes and added references

References in corpus (6)

• Two Dimensional Gauge Theories Revisited
• Toric reduction and a conjecture of Batyrev and Materov
• Residues formulae for volumes and Ehrhart polynomials of convex polytopes
• Vector partition function and representation theory
• The volume of the 10th Birkhoff polytope
• Witten's Nonabelian Localization for Noncompact Hamiltonian Spaces

Cited by in corpus (18)

• A fixed point formula for the index of multi-centered N=2 black holes
• Geometric quantization and families of inner products
• On geometric quantization of $b$-symplectic manifolds
• Index of transversally elliptic operators
• Lecture Notes on Equivariant Cohomology
• Equivariant relative Thom forms and Chern characters
• Equivariant Chern characters with generalized coefficients
• Torus fibrations and localization of index III
• A K-homological approach to the quantization commutes with reduction problem
• A geometric realisation of tempered representations restricted to maximal compact subgroups
• Hirzebruch class and Bialynicki-Birula decomposition
• $S^1$-equivariant local index and transverse index for non-compact symplectic manifolds
• An equivariant index formula for elliptic actions on contact manifolds
• Formal geometric quantization II
• A Danilov-type formula for toric origami manifolds via localization of index
• Shifts of prepotentials (with an appendix by Michele Vergne)
• A geometric formula for multiplicities of $K$-types of tempered representations
• Background cohomology of a non-compact Kahler G-manifold