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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.

Course Hours
Type
Day
Hours
Format
Lecturer
Lecture
Friday
1500 - 1630
online
Prof. Dr. Peter Bürgisser

Due to the current situation, the course will be given online via Zoom.
Complementary material accompanying the lecture 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

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

Part II. Moment map and moment polytopes

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

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

Zusatzinformationen / Extras

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