**Sushmita Venugopalan**

TIFR

October 18, 2012

**Moment maps, Yang-Mills functional and vortices**:
Consider a complex reductive group acting linearly on a non-singular
variety. According to the work of Kirwan, the semistable locus can be
considered the open stratum in the Morse stratification induced by the
square of the moment map. If the symplectic quotient exists, it coincides with the GIT quotient.
The Narasimhan-Seshadri-Donaldson theorem is an infinite dimensional version of this phenomenon, with the moment map being the Yang-Mills functional. This theorem relates the space of semi-stable holomorphic structures on a vector bundle to the space of Hermitian-Einstein connections.
Similar results have also been proved on the space of holomorphic bundles with some additional data, for example a holomorphic section. In this case, setting the moment map equal to zero gives the vortex equation.