The symplectic vortex equations and applications

辛涡旋方程是最近由Cieliebak，Gaio，Mundet和Salamon提出的具有Hamilton群作用的辛流形上的方程。在过去的几年里，我们发展了这些方程的解的理论。在这个项目中，我们将应用辛涡方程的问题，在全球微分几何。主要应用是枚举几何，扩展工作Kontsevich Manin的Gromov-Witten不变量和Givental镜像对称。其他应用关注维滕的猜想上的Verlinde代数，和不同的规范之间的关系理论不变量的光滑四流形。

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.