We discuss the bound state spectrum of a particle moving on a Riemannian manifold under the influence of delta function potentials. We show that the renormalization can be naturally accomplished by the heat kernel approach. A slight generalization to the Klein-Gordon type particles are also presented. Currently there is some interest on particles captured by curves, modeled by singular interactions supported on such curves, we present a generalization to our approach to circles embedded into a Riemannian manifold. We also discuss the renormalization group flows of these systems. If time permits a more ambitious generalization to interacting bosons on a two dimensional manifold will be presented.

**Venue:** Department of Mathematics, Trieste University, Seminar Room - Third Floor bld H2bis