Laplacians and Spectrum for Singular Foliations

The author surveys Connes' results on the longitudinal Laplace operator along a (regular) foliation and its spectrum, and discusses their generalization to any singular foliation on a compact manifold. Namely, it is proved that the Laplacian of a singular foliation is an essentially self-adjoint operator (unbounded) and has the same spectrum in every (faithful) representation, in particular, in L-2 of the manifold and L-2 of a leaf. The author also discusses briefly the relation of the Baum-Connes assembly map with the calculation of the spectrum.