ABSTRACT: On February 5, I discussed the problem of understanding the representation theory of Hecke correspondences between moduli spaces of stable sheaves on a smooth projective surface S (and presented reasons why one would care, stemming from mathematical physics and algebraic geometry). In this talk, I will give details on the proof, by studying the geometry and intesection theory of various nested moduli spaces. This will be essentially an algebraic geometry talk, but I will emphasize the big picture and future perspectives.