Gromov-Witten Invariants and Symplectic Vortices

In 1985, Gromov introduced the notion of J-holomorphic curves into symplectic geometry, for which they have proved to be a powerful tool. The Gromov-Witten invariants of a symplectic manifold (M, !) are given by the number of J-holomorphic curves that represent a fixed homology class and at fixed or varying points pass through given submanifolds of M. Here J is an !-compatible almost complex structure. The symplectic vortex invariants on the other hand are associated to a symplectic manifold (M, !) with a Lie group G acting on M in a Hamiltonian way. In this overview article I will describe both invariants and I will explain what the relation between the two is.