Tuesday, September 13, 2022, 03:45pm - 04:45pm
Let's says you have a group G acting on a variety (or scheme) X and you want to take the quotient. Intuitively you may want to take some version of the "orbit space" but it turns out there is not always an obvious way to put a scheme structure on it. Geometric invariant theory (GIT) gives us a way to solve this problem. The simplest solution is given by the affine GIT quotient, which is given by taking Spec of G-invariant functions on X. If X is an affine variety, points in the affine quotient will turn out to correspond to closed orbits in X. In particular, if G is finite we recover the orbit space as a scheme, although in general the behavior can be pathological. Thus sometimes a more sophisticated answer is needed, which is given by the projective GIT quotient. We shall study both of these and their relationship, with one key example being Kleinian singularities (quotients of A^2 by finite groups). Time permitting we will briefly talk about Hamiltonian reductions and Nakajima quiver varieties. No background will be assumed.
Location: 12th floor classroom