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Vorlesung Fortgeschrittene Themen der Algebra (2h):

Group actions, representations and geometric invariant theory 

LV-Nummer: 3236 L 6109

Fr 10:00-11:30 MA 376

Begin: 17. April

The goal of this course is to present the part of my book draft
"Optimization, Invariants, and Complexity", which develops the algebraic foundations,
and which is almost finished. 

A pdf containing the corresponding book chapters will be made available to the participants
(around 60 pages).

I hope to present the geometric and algorithmic aspects of the theory in a follow up course.
(For this, the book draft still needs considerable work.)

The course is written in a way to keep the prerequisites to a minimum. Still many concepts come
into play. A basic knowledge of group representation theory (as in my course in the summer semester  2025)
and some concrete understanding of basic concepts of algebraic geometry (as developed in Dominic Bunnett's course
in the winter semester  2024/25) will be helpful.

The course will also nicely complement the planned student seminar on algebraic groups with Dominic Bunnett
in the summer semester  2025.

Provisional Table of Content

Groups, Actions, Invariants
    Self-adjoint groups 
    Structure theorems for groups    
    Representations 
    Reynolds operator 
    Hilbert’s finiteness theorem 
    Orbits and their closures 
    Weight space decomposition 

Geometric Invariant Theory
    Hilbert-Mumford criterion 
    Generalized Hilbert-Mumford criterion 
    Convexity, moment map, and Kempf-Ness theorem

Representations and Highest Weights 
    Highest weights and Weyl chambers 
    Schur-Weyl duality 
    Algebraic Peter-Weyl theorem 

If there is time:

Moment Polytopes 
       Moment map for torus actions 
       Coadjoint orbits 
       Moment polytopes