  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.
