9704774 Givental This research project lies at the interface of topology, geometry, and mathematical physics. The investigator has recently given a proof of the mirror symmetry conjecture for a class of algebraic varieties, including the classical case of quintics in four dimensional complex projective space. The current project is a continuation of that work and pursues several new directions in quantum cohomology theory. Mirror symmetry is a phenomenon first uncovered by physicists in connection with the popular 10-dimensional string model of the universe. In this model, the six invisible dimensions arise as so called Calabi-Yau manifolds; they observed that a model of conformal field theory known as the A-model of a Calabi-Yau manifold coincides with another conformal field theory model, the B-model, of its 'mirror' manifold. In mathematical terms this means that there is a well-defined correspondence between the symplectic topology of a Calabi-Yau manifold and the complex geometry of its mirror manifold.
