Recursive formulas for Gromov-Witten invariants

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Award: DMS-0071393Principal Investigator: Eleny-Nicoleta IonelThe goal of the first project is to obtain new relations in thecohomology of the moduli space of complex structures on a markedRiemann surface. Using techniques developed in her earlier work,the PI found several interesting relations, one of which couldprove the Faber conjecture about the generators of thetautological ring. The second project seeks to find relationsbetween the relative and absolute Gromov-Witten invariants. Suchrelations appear to be useful in mirror conjecture computations,as well as in several other open problems in enumerativegeometry. The final project suggests two ways of extending theGromov-type invariants to `nongeneric' situations. This wouldprovide more refined information about the symplectic manifold. Most of the problems in enumerative algebraic geometry are morethan a hundred years old. The questions are easy to ask, but theprogress in solving them using classical methods has been quiteslow. Recently, the same kind of questions arised in twodimensional topological quantuum field theories from high energyphysics. Inspired by these theories, new methods lead in the pastcouple of years to amazing progress in the field. The proposalexplores two new ways of approaching these old problems thatwould further clarify the structure of the two dimensionaltopological quantuum field theories.