Schönberg Correspondences and Cohomology

By Schönberg correspondence we mean the correspondence between positive-type (suitably continuous) semigroups and their infinitesimal generators. The classical Schönberg correspodence is for generators of semigroups of positive definite kernels. Many more instances are known. A Schönberg correspondence alway includes a precise characterization of the possible generators, usually including some condition which is called conditional positivity. Frequently, finding all generators involves the solution of cohomologic problems. This is what we will to speak about.