# Tom Baird

## Topological Aspects Of GIT Over The Real And Complex Numbers

Date Monday, October 24, 2016

When a reductive group G acts on a polarized projective variety $X$, Mumford's geometric invariant theory defines a quotient variety $X//G$. This construction is very simple to define algebraically, but the geometric relationship between $X$ and $X//G$ is rather subtle. Over the complex numbers, the Kempf-Ness theorem provides an illuminating geometric construction of $X//G$ using Morse theory. I will give an overview of these ideas. If time permits, I will describe some recent work with my student Nasser Heydari proving analogues of these results over the real numbers.