Evaluating tautological classes using only Hurwitz numbers -- Renzo Cavalieri

Hurwitz numbers count ramified covers of a Riemann surface with prescribed monodromy. As such, they are purely combinatorial objects. Tautological classes, on the other hand, are distinguished classes in the intersection ring of the moduli spaces of Riemann surfaces of a given genus, and are thus ``geometric.'' Localization computations in Gromov-Witten theory provide non-obvious relations between the two. In this talk we will illustrate these ideas - in particular we'll give an elementary proof of a classical computation by Faber-Pandharipande, expressing in generating function form the evaluation of the hyperlliptic loci in the tautological ring(s) R^star(M_g) (for all g). Besides knowing how to localize, all we need is the number 1/2, in its multiple incarnations.