Recursive formulas for Gromov-Witten invariants

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

Award: DMS-0071393 Principal Investigator: Eleny-Nicoleta Ionel The goal of the first project is to obtain new relations in the cohomology of the moduli space of complex structures on a marked Riemann surface. Using techniques developed in her earlier work, the PI found several interesting relations, one of which could prove the Faber conjecture about the generators of the tautological ring. The second project seeks to find relations between the relative and absolute Gromov-Witten invariants. Such relations appear to be useful in mirror conjecture computations, as well as in several other open problems in enumerative geometry. The final project suggests two ways of extending the Gromov-type invariants to `nongeneric' situations. This would provide more refined information about the symplectic manifold. Most of the problems in enumerative algebraic geometry are more than a hundred years old. The questions are easy to ask, but the progress in solving them using classical methods has been quite slow. Recently, the same kind of questions arised in two dimensional topological quantuum field theories from high energy physics. Inspired by these theories, new methods lead in the past couple of years to amazing progress in the field. The proposal explores two new ways of approaching these old problems that would further clarify the structure of the two dimensional topological quantuum field theories.