This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology QHG(X) of a smooth polarized complex projective variety X with the action of a connected complex reductive group G to the orbifold quantum cohomology QH(X//G) of its geometric invariant theory quotient X//G, and prove that it intertwines the genus zero gaug...

We describe the structure of mirror formulas for genus 0 Gromov-Witten invariants of projective complete intersections with any number of marked points and provide an explicit algorithm for obtaining the relevant structure coefficients. As an application, we give explicit closed formulas for the genus 0 Gromov-Witten invariants of Calabi-Yau complete intersections with 3 and 4 constraints. The ...

We present a uniform version of a result of M. Gromov on the surjunctivity of maps commuting with expansive group actions and discuss several applications.

It is known that the Cayley graph ? of a negatively curved (Gromov-hyperbolic) group G has a well-deened boundary at innnity @?. Furthermore, @? is compact and metrizable.

Introduction Gromov-Witten invariants are enumerative invariants of stable maps. Their definition in the context of mirror symmetry in physics allowed new approaches to old problems — for instance, counting the number of plane rational curves of degree d through 3d − 1 points — and solved all at once enumerative problems that had thwarted mathematicians for years. Any such innovation gives rise...