Abstract:A Hamiltonian action of a Lie group on a symplectic manifold $(M,\omega)$ gives rise to a gauge theoretic deformation of the Cauchy-Riemann equations, which are called the symplectic vortex equations. I will explain how counting solutions of these equations over the complex plane leads to a ring homomorphism from the equivariant cohomology of $M$ to the quantum cohomology of the symplectic quotient, assuming that $M$ is symplectically aspherical.