The symplectic vortex equations and applications

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The symplectic vortex equations are equations on a symplectic manifold with a Hamiltonian group action recently introduced by Cieliebak, Gaio, Mundet and Salamon. Over the past years we developed the solution theory of these equations. In this project we will apply the symplectic vortex equations to questions in global differential geometry. The main application is to enumerative geometry, extending work of Kontsevich-Manin on Gromov-Witten invariants and Givental on mirror symmetry. Other applications concern Witten's conjecture on the Verlinde algebra, and the relation between different gauge theoretical invariants of smooth four-manifolds.