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Regular version of the site
2020/2021

## Invariant Theory

Type: Optional course (faculty)
When: 3, 4 module
Language: English
ECTS credits: 6

### Course Syllabus

#### Abstract

The classical Invariant Theory studies the rings of functions on a vector space invariant with respect to an action of analgebraic group. One of the standard problems is to find a nice set of generators and relations of this ring. More invariantly, the aim is to define and compute the quotient of an algebraic variety by anaction of analgebraic group (roughly parametrizing the set of orbits). This theory originatesin 19th century and is one the most efficient tools of constructing and studying the new algebraic varieties (forexample, the quiver varieties). Рrerequisites: First two years of our program: Linear algebra, basica lgebra,complex analysis, analysison manifolds.

#### Learning Objectives

• 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.

#### Expected Learning Outcomes

• 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

#### Course Contents

• Invariants under reductive group actions
We 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 goals
We give an introduction to the origins and goals of Invariant Theory
• Birational invariants
We prove separation of generic orbits by birational invariants.
• Invariant theory of finite groups
We 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 closures
We 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

#### Assessment Elements

• home assignments
• final exam

#### Interim Assessment

• Interim assessment (4 module)
0.3 * final exam + 0.7 * home assignments

#### Recommended Core Bibliography

• Dolgachev, I. (2003). Lectures on Invariant Theory. Cambridge University Press.