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# Selected topics in invariant and representation theory

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

Prerequisites: I will assume a rough familiarity with matrix Lie groups, as explained in my lecture Geometric Invariant Theory in summer 2020. This roughly covers Part I in Brian Hall's textbook.

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

## Schedule and Organization

Please note that the lecture starts at ~~12:00~~ 15:00 **sharp**. *(NEW starting time)*

The course starts on April 16.

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

Lecture | Friday | 15^{00} - 16^{30} | online | Prof. Dr. Peter Bürgisser |

Due to the current situation, the course will be given ** online via Zoom**.

**accompanying the lecture**

*Complementary material***.**

*will be provided on ISIS***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).

## Goals of the lecture

1. Introduction to the representation theory of semisimple Lie algebras

2. Discussion of the moment map of a complex representation of a reductive group, convexity theorem (moment polytope)

The aim is to provide some of the mathematical background of the recent paper *`Towards a theory of non-commutative optimization: geodesic first and second order methods for moment maps and polytopes'* by Bürgisser, Franks, Garg, Oliveira, Walter, and Wigderson, see arXiv:1910.12375.

The motivation from this research comes from geometric complexity theory and a new research direction dealing with optimization on (non-commutative) groups.

## Current planned content

**Part I. Representations of semisimple Lie algebras**

Lie Groups, Lie Algebras and Representations by Brian Hall (Springer GTM 222), Chapters 5-7*Main source:*Roots, dominant weights, theorem of highest weight, constructions of representations, Weyl's character formula**Topics:**

**Part II. Moment map and moment polytopes**

several research papers. More information will be provided later.*Sources:*Convexity theorem, toric case, Applications (Schur-Horn, Horn's problem)*Topics:*

## Literature

- Hall,
*Lie Groups, Lie Algebras and Representations*, Springer GTM 222 - Fulton and Harris,
*Representation Theory*, Springer - Hoskins,
*Geometric Invariant Theory and Symplectic 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