- The goal of this course is to study the invariants of the actions of algebraic groups on algebraic varieties (for example, on vector spaces). We will prove the classical foundational theorems of Hilbert, Weyl and Mumford. We will also prove the more recent theorems of Luna, Vinberg, Deligne and others.
- The students will learn the origins and goals of the classical Invariant Theory
- The students will learn the proof of Chevalley-Shephard-Todd theorem
- The students will learn how to separate generic orbits by birational ivariants
- The students learn the basic notions of reductive groups' theory
- The students will learn the proof of the Hilbert-Mumford theorem and of Kempf's theorem of optimal destabilizing subgroups.
- The students will learn connection to moment maps for compact group actions (the Kempf-Ness theorem) and its applications.
- The students will learn the proof of the fundamental theorems of the classical invariant theory
- Invariants under reductive group actionsWe give a brief review of reductive groups, Hilbert's theorems on quotients and quotient morphisms. Algebro-geometric properties of quotients.
- A brief introduction to Invariant Theory: its origins and goalsWe give an introduction to the origins and goals of Invariant Theory
- Birational invariantsWe prove separation of generic orbits by birational invariants.
- Invariant theory of finite groupsWe prove the Chevalley-Shephard-Todd theorem
- Computation of invariants: Classical invariant theory.We prove the fundamental theorems of the classical invariant theory
- Closed orbits in orbit closuresWe prove the Hilbert-Mumford theorem, Kempf's theorem of optimal destabilizing subgroups.
- Connection to moment maps for compact group actions (the Kempf-Ness theorem) and applications.We prove the Kempf-Ness theorem