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 L2 of the manifold and L2 of a leaf.The author also discusses briefly the relation of the Baum-Connes assembly map with the calculation of the spectrum.