TU Berlin

Fachgebiet Algorithmische AlgebraInvariant Theory

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Invariant Theory: Computational and Geometric Aspects

Schedule

There will be no lectures in the weeks June 4-8 and June 18-22 !! 

Replacement dates:

  • Tuesday, July 10: 830 - 1000
  • Wednesday, July 11: 1600 - 1730
Vorlesungszeiten
Typ
Tag
Zeitraum
Raum
Lehrperson
Vorlesung
Tuesday
1015-1145
MA 316
Prof. Dr. Peter Bürgisser
Vorlesung
Wednesday
1415-1545
MA 316
Prof. Dr. Peter Bürgisser
Übung
Friday
1015-1145
MA 316
Kathlén Kohn

Exercise Sheets

There are exercise sheets available which are not graded or marked:

Content

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 invariant theory, and to prepare students to understand some of the recent work relating complexity and invariant theory. Students that intend to write a thesis under the direction of Prof. Bürgisser are expected to attend this lecture.

The prerequisites of the course are a basic knowledge in algebraic geometry (e.g., lecture algebraic geometry I). Some familiarity with concepts of representation theory is useful, but not necessary. 

Table of Content

Keywords

  • Symbolic Method (Polarization)
  • Cayley's Omega Process
  • Fundamental Theorem of Invariant Theory
  • Reductive groups
  • Quotients
  • Hilbert-Mumford criterion
  • Invariants of finite groups
  • Invariants of GL(V)

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

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