Gromov - Witten invariants and integrable hierarchies

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AbstractAward: DMS-0306316Principal Investigator: Alexander GiventalGromov-Witten invariants characterize global properties of phasespaces in Hamiltonian mechanics. The project concerns withvarious problems of constructing and generalizing Gromov-Witteninvariants, developing methods of their computation, interpretingthe geometrical information they encode and understanding therules that govern relationships among them. In particular, therole of integrable hierarchies and their Virasoro symmetries inGromov-Witten theory and in its mirror partner - singularitytheory - is to be investigated. To this end, certain vertexoperator constuctions based on the theory of vanishing cycles insingularity theory will be explored as a promising tool fordescribing and manipulating with integrablehierarchies. Exploiting the twisted loop group of hidensymmetries discovered recently in the structure of Gromov-Witteninvariants and uncovering the origin of the symmetries are amongkey motives for this project.In a broader prospective, the projects continues the legacy ofthe past two centures in the development and application ofmodern mathematics. The methods of algebraic geometry, topology,group theory and quantum mechanics laid down in the classicalwork of Gauss, Riemann, Lie, Klein, Poincare and Weyl cometogether to address the problems posed lately by string theory inits search for the ultimate unity of fundamental interactions.