Equivariant quantum cohomology of homogeneous spaces.

The (small) equivariant quantum cohomology algebra of a homogeneous space X = G/P is a deformation of both equivariant and quantum cohomologies of X. It has an additive basis determined by Schubert classes. Its multiplicative structure constants are the (3-point, genus 0) equivariant Gromov-Witten invariants, considered first by A. Givental and B. Kim. The usual Gromov-Witten invariants and the structure constants of equivariant cohomology of X form a subset of these invariants. Those in the complement of this subset are called mixed. The purpose of this thesis is to study the equivariant quantum cohomology of G/P and in particular the equivariant Gromov-Witten invariants. We prove first an equivariant quantum Chevalley rule, i.e. a rule to multiply a Schubert class with a divisor Schubert class. It turns out that this rule is the simplest possible: it doesn't contain any mixed terms. In general, the divisor classes do not generate the equivariant quantum cohomology algebra. In spite of this, we give an effective algorithm to compute the equivariant Gromov-Witten invariants. The main tool used in this algorithm is a recurrence formula derived from the equivariant quantum Chevalley rule. We also prove that the invariants in question satisfy the same positivity property as the one in equivariant cohomology. They are homogeneous polynomials in the variables associated to negative simple roots with nonnegative coefficients. In the equivariant context, this was conjectured by D. Peterson and proved by W. Graham. Finally, we study further the case when X is a Grassmannian. In this case we obtain a presentation of the equivariant quantum cohomology by generators and relations. We show that with respect to this presentation, the Schubert classes are represented by certain determinants associated to a Jacobi-Trudi formula for the factorial Schur functions.Ph.D.MathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/125164/2/3186707.pd