Vergne, Michèle 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]. Cited in 2 ReviewsCited in 22 Documents 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 Keywords:Hamiltonian action; symplectic reduction; localization formula; polytope; index; transversally elliptic operator; spline; Euler-Maclaurin formula PDFBibTeX XMLCite \textit{M. Vergne}, in: 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). 635--664 (2007; Zbl 1123.19004) Full Text: arXiv