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4:00 pm Tuesday, December 3, 2013 Algebraic Geometry Seminar: The Gromov Witten/Hurwitz correspondence (after Okounkov and Pandharipande)by Kuan-Wen Lai (Rice) in HB 227- Hurwitz number counts the number of coverings over a Riemann surface with specific ramification conditions. It has a representation-theoretical reinterpretation, which allows it to be expressed by a classical basis in the space of shifted symmetric functions. The GW/H correspondence could be easily constructed on the smooth locus of the moduli space of stable maps. However, a full correspondence requires a completion of the classical basis, called completed cycles, to cover the boundary terms. Completed cycles are actually the eigenvalues of some special operators in the infinite wedge space, which in term introduces the operation formalism into the derivation of the correspondence. GW/H correspondence is a conclusion of its special case while the target space is the projective line. In the special case, the GW-invariants and the Hodge integrals could be related in the form of generating functions. Using the ELSV formula, this equation could be transformed into an operator formula, which in term gives us the special correspondence. Finally, the degeneration formula decomposes the GW-invariants into a deviated form of the full correspondence. Here the special correspondence would indicate that the expression is actully the same as the required one.
Submitted by btl1@rice.edu |