A surprising connection between symplectic geometry and geometric invariant theory

Abstract: 

In the 1980s, a relationship was found between the two fields of symplectic geometry and geometric invariant theory (GIT) via the Kempf-Ness theorem. Symplectic geometry generalises the notion of the phase space in classical mechanics, while GIT studies quotients by group actions in algebraic geometry.

In this talk, we give a brief overview of both symplectic geometry and GIT with the intent to discuss a corollary of the Kempf-Ness theorem. The corollary is the following: Given a smooth complex projective variety X and a complex reductive Lie group G, the GIT quotient of X by G is homeomorphic to the symplectic reduction of X by a maximal compact subgroup of G.