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# Geometric Invariant Theory

- Lecture: 2h, 5 ECTS
- Eligible as BMS Advance Course in Area 2
- Anrechenbar als Modul „Fortgeschrittene Themen der Algebra”

The prerequisites of the course are a basic knowledge in algebraic geometry (e.g., lecture algebraic geometry I).

## Schedule and Organization

Please note that the lecture starts 10am **sharp**.

Type | Day | Hours | Format | Lecturer |
---|---|---|---|---|

Lecture | Tuesday | 10^{00} - 11^{30} | online | Prof. Dr. Peter Bürgisser |

Due to the current situation, the course will be given **online**. Complementary material accompanying the lecture will be provided. This will be a new experience for all of us: we have to see how well it goes! To start with, we try to keep the schedule as announced in the Vorlesungsverzeichnis.

**People interested to participate** in the course are asked to **send an email** to Peter Bürgisser (pbuerg at math.tu-berlin.de) with cc to Philipp Reichenbach (reichenbach at tu-berlin.de), and to register at the corresponding ISIS course website (NEW).

Shortly before the first meeting on April 21, you will then obtain an invitation along with further instructions.

I expect people in my group to participate in this course: it provides the necessary background for an exciting research project we are currently untertaking.

## Description

Invariant theory is a classical area of mathematics that played a central role in the development of algebra, even though many of its concrete and computational results from the 19th century almost fell into oblivion. Invariant theory got new impetus with Mumford's introduction of geometric techniques. Recently, it was discovered that quantitative and algorithmic questions of classical invariant theory are closely related to fundamental questions of algebraic complexity theory. This led to surprising progress for computational questions.

The goal of the lecture is to give an introduction to geometric invariant theory, following the recent textbook *Geometric Invariant Theory: over the real and complex numbers* by Nolan R. Wallach (Springer Universitext 2017).

## Topics

- Basics on Lie groups and Lie algebras
- Reynolds operator and Hilbert's finiteness theorem for invariants
- Invariants and closed orbits
- Hilbert-Mumford Theorem
- Kempf-Ness-Theorem
- Some applications (e.g. matrix scaling, operator scaling)

## Literature

- Derksen and Kemper,
*Computational Invariant Theory*, Springer - Dieudonne and Carrel,
*Invariant Theory, Old and New*, Academic Press - Dolgachev,
*Lectures on Invariant Theory*, Cambridge Univ. Press - Hoskins,
*Geometric Invariant Theory and Syplectic Quotients*, lectures notes FU Berlin - Kraft,
*Geometrische Methoden in der Invariantentheorie*, Vieweg - Mumford, Fogarty, Kirwan,
*Geometric Invariant Theory*, Springer - Newstead,
*Introduction to Moduli Problems and Orbit Spaces*, lecture notes, TIFR - Procesi,
*Lie Groups: An Approach through Invariants and Representations*, Springer - Sturmfels,
*Algorithms in Invariant Theory*, Springer - Wallach,
*Geometric Invariant Theory: Over the Real and Complex Numbers*, Springer Universitext