MATH 761, Invariant theory (Spring 2025).

Instructor: Prof. Ivan Loseu (email: ivan.loseu@yale.edu)

Lectures: MW 2:30-3:45pm, location: KT 207. The first class is on 1/13

Office hours: T 2.30-3.30, W 10.30-11.30.

This document contains important information on the class content. Please read it!

This highly invariant bear seen by Tianqi Wang is roaming Yale Math department looking for homework submissions!

Assignments:

  • Homework 1, due Feb 5.
  • Homework 2, due Feb 21.
  • Homework 3, due Mar 28.
  • Homework 4, due Apr 17.

    Schedule:

  • January 13, Lec 1: Introduction, notes. I plan to explain a motivation coming from Linear algebra, have a brief discussion of polynomial invariants and state foundational results of Hilbert. Finally, we mention applications to Algebraic geometry.
  • January 15, Lec 2: Categorical quotients, I, notes. We discuss averaging operators in a general setting, apply them to prove the Hilbert finiteness theorem, and discuss important special cases.
  • January 20: MLK, no class.
  • January 22, Lec 3: Categorical quotients, II, notes. Actions on affine varieties and categorical quotients.
  • January 24, Lec 4: Categorical quotients, III, notes (updated Jan 29). We discuss algebro-geometric properties of quotients as well as reductive groups.
  • January 27, Lec 5: \theta-groups I, notes. This is a class of linear actions of reductive groups with very nice properties. We will specify this class, describe motivations and start developing the structure theory.
  • January 29, Lec 6: \theta-groups II, notes. We discuss Cartan subspaces and Weyl groups in the \theta-group setting.
  • February 3, Lec 7: \theta-groups III, notes. We prove the Chevalley restriction theorem for \theta-groups and (at least) start proving that the Weyl group is a complex reflection group.
  • February 5, Lec 8: \theta-groups IV, notes (updated 2/6). We finish the proof that the Weyl group is a complex reflection groups and start a discussion of computing \theta-groups.
  • February 10, Lec 9: \theta-groups V, notes (updated 2/14). We continue our discussion of computing the \theta-groups.
  • February 12, Lec 10: \theta-groups, VI, notes. We finish the discussion of SL_3 actiong on S^3(C^3) and discuss sections of the quotient morphisms. Here is a complete proof of the main theorem from Section 2.2.
  • February 17, Lec 11: Hilbert-Mumford theorem, I, notes. We state and prove the theorem about detecting closed orbits in the closure of a given one using one-parameter subgroups.
  • February 19, Lec 12: Hilbert-Mumford theorem, II, notes. We do computations using the Hilbert-Mumford theorem, and start discussing optimal destabilizing subgroups.
  • February 24, Lec 13: Hilbert-Mumford, III, notes. We continue the study of characteristics/ optimal destabilizing subgroups and introduce the Hesselinl/Kirwan-Ness stratification of the zero fiber of the quotient morphism.
  • February 26, Lec 14: Etale slices, notes. We discuss the etale slice theorem due to Luna. Proof of the main lemma.
  • March 3, Lec 15: Connections to Symplectic geometry, I notes. We state and prove the Kempf-Ness theorem, discuss moment maps and deduce some applications.
  • March 5, Lec 16: Connections to Symplectic geometry, II notes. We discuss Hamiltonian reduction and symplectic slice theorems. The participants may want to familiarize themselves with cotangent bundles (Sec 2 in [CdS]) and local structure theorems (Part III in [CdS], especially Secs. 6 and 7).
  • Bonus for lecture 16.
  • March 24, Lec 17: GIT quotients, I, notes. We start our discussion of GIT quotients.
  • March 26, Lec 18: GIT quotients, II, notes. We discuss the Hilbert-Mumford criterium for semistability and give some examples of computation of semistable loci and GIT quotients.
  • March 31, Lecture 19: GIT quotients, III, notes. GIT Hamiltonian reductions: Hilbert schemes of points on the plane and Calogero-Moser spaces.
  • April 2, Lec 20: GIT quotients, IV notes. We discuss the Kempf-Ness type theorem for GIT quotients, as well as the hyper-Kahler reductions.
  • April 7, Lecture 21: Moduli spaces, I notes. We start discussing moduli spaces of vector bundles on a smooth projective curve and gather relevant facts and constructions from Algebraic geometry.
  • April 9, Lecture 22: Moduli spaces, II, notes. We continue our discussion of Quot schemes and discuss GIT for actions on projective schemes.
  • April 14, Lecture 23: Moduli spaces, III, notes. We discuss the GIT of the Quot scheme.
  • April 16, Lecture 24: Moduli spaces, IV. We finish our discussion of moduli spaces of (semi)stable bundles by relating the usual (semi)stability to the GIT (semi)stability.