Two talks between 14:00 and 16:30, at Lecture Hall 1, with a teabreak in between (1) Dynamics on the moduli space of pointed rational curves (2) A mirror theorem for symmetric products of projective space (1) Rohini Ramadas (2) Robert Silversmith Ann Arbor. 040117 Abstract (1) The moduli space M_{0,n} parametrizes all ways of labeling n distinct points on P^1, up to projective equivalence. Let H be a Hurwitz space parametrizing holomorphic maps, with prescribed branching, from one nmarked P^1 to another. H admits two different maps to M_{0,n}: a ``target curve'' map pi_t and a ``source curve'' map pi_s. Since pi_t is a covering map, pi_s(pi_t^(1)) is a multivalued map  a Hurwitz correspondence  from M_{0,n} to itself. Hurwitz correspondences arise in topology and Teichmuller theory through Thurston's topological characterization of rational functions on P^1. I will discuss their dynamics via numerical invariants called dynamical degrees. (2) Through 3 general points and 6 general lines in P^3, there are exactly 190 twisted cubics; 190 is a GromovWitten invariant of P^3. Mirror symmetry is a conjecture about the structure of all GromovWitten invariants of a smooth complex variety (or orbifold) X. The conjecture is known for toric orbifolds and some of their complete intersections. We prove it in the case of the nontoric orbifold Sym^d(P^r). This orbifold is of particular interest because when r=2, its GromovWitten invariants are conjecturally related to those of the Hilbert scheme Hilb^d(P^2).
