Geometric Invariant Theory

Geometric invariant theory is the study of the action of algebraic groups on schemes, and the construction of the associated quotient, or orbit spaces. In this reading group, we will first define the basic notions of (reductive) algebraic groups, and GIT applied to affine and projective schemes.

A moduli problem in algebraic geometry involves classifying geometric objects up to isomorphism. Often, this involves defining a larger parameter space, and a group acting on it, where isomorphic objects lie in the same orbit. We can then use GIT to construct the associated moduli space. In particular, this can be applied to the moduli space of curves, the moduli space of projective hypersurfaces and the moduli of vector and principle bundles on curves. The main goal of the reading group is to study applications of GIT to moduli problems.

Finally, when we are working with quasi-projective varieties over the complex numbers, there is another notion of quotients, coming from symplectic geometry. The Kempf-Ness theorem relates the two quotient constructions, and there are many relations to (for
example) gauge theory and representation theory.

Organizer: Shing Tak-Lam