Abstract: This will be a survey talk. Let R be a Riemann surface, I will talk about some spaces of automorphic functions on R. Poincare series map is one way to construct automorphic forms, but it might not be convergent when we have an automorphic function. Some projections on automorphic functions will be introduced, also I will show few applications of these projections.

Abstract: This time we look at the cotangent bundle of a representation M of a torus. We discuss how the infinitesimal index gives an isomorphism between equivariant cohomology modules induced by the orbit filtration of M and certain spaces of splines which we are going to define.

Abstract: We give an introduction to the theory of local cohomology by pointing out examples and its connections to topology, differential equations, and algebraic geometry. We discuss, in the presence of a field, some techniques for investigating local cohomology modules, focussing on the study of finiteness conditions. We show how the absence of a field, or singularities in the ambient space, complicate things significantly, and work out the case of the variety consisting of matrices of bounded rank over every field and over the integers.