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Applications of equivariant cohomology. (English) Zbl 1123.19004

Sanz-Solé, Marta (ed.) et al., Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume I: Plenary lectures and ceremonies. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-022-7/hbk). 635-664 (2007).
Summary: 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.
For the entire collection see [Zbl 1111.00009].

MSC:

53D20 Momentum maps; symplectic reduction
55N25 Homology with local coefficients, equivariant cohomology
58J20 Index theory and related fixed-point theorems on manifolds
19L47 Equivariant \(K\)-theory
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