Gromov-Witten invariants and symplectic reduction

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AbstractAward: DMS-0604705Principal Investigator: Alexander GiventalGromov -- Witten invariants characterize global properties ofphase spaces in Hamiltonian mechanics. It is our understandingthat Gromov--Witten theory has entered a very productive periodof its development. It is about to become a part of a moregeneral framework based on ideas of conformal field theory. Manykey outstanding problems of the theory such as, e.g. Virasoroconjecture, or Witten's W_n-gravity conjecture are likely toget either resolved on the basis of this framework or at leastadvanced to a substantial degree. Respectively, we are planningto continue our study of GW-invariants, their axiomaticstructure, their generalizations, their relationships withintegrable systems and symplectic field theory, methods of theircomputation including those associated with the mirrorconjecture, and try to utilize the advantages that may come fromconformal field theory. A particular problem that gave theproposal its name is a long-standing conjecture about thebehavior of Gromov-Witten invariants under symplectic reductionof target spaces by circle actions.From a more general prospective, problems we deal with in ourresearch lie on the crossroad of two major pathways inmathematics of the last two centuries. One of them is thein-depth pursuit of the intricate properties of algebraic curves--- in the form inherited from works of Gauss, Abel, Jacobi,Riemann, Klein and Poincare. The other is the broad conceptuallandscaping of mathematical physics dictated by the progress ofclassical and quantum mechanics and often associated with thenames of Hamilton, Maxwell, Gibbs, Poincare, Hilbert, Einsteinand Weyl. It is string theory that in the search for theultimate laws of nature places algebraic curves in the center ofthe modern landscape of fundamental physics, and generates newmathematical questions and points out plausible answers with anamazing pace and persistence. Some of the problems we work onare motivated by such questions, some others hopefully providethe answers that string theory did not really anticipate.