Advisor:  Dr. Mark ShoemakerCo-Advisor:Dr. Renzo Cavalieri

Committee:  Dr. Alexander Hulpke, Dr. Chris Peterson, Dr. Maria
Gillespie, Dr Hua Chen

Title: A Quantum H∗(G)-module via Quasimap Invariants

Abstract: For X a smooth variety or Deligne—Mumford stack, the quantum
cohomology ring QH^*(X) is a deformation of the usual cohomology ring
H^*(X), where the product structure is modified to incorporate quantum
corrections.  These correction terms are defined using Gromov—Witten
invariants.   For a GIT quotient V//G, the cohomology ring H^*(V//G)
also has the structure of a H^*(G)-module.  In this work, we
use quasimap invariants with light points and a modified version of the
WDVV equation to define a quantum deformation of this H^*(G)-module
structure.  Using localization, we explicitly compute this structure for
the Hirzebruch surface of type 2.  We conjecture that this new quantum
module structure is isomorphic to the Batyrev ring when the target is a
semipositive toric variety.

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