Forschungsgruppe Nichtlineare Algebra

Direktor:
Bernd Sturmfels

Kontakt: Email
Telefon:
+49 (0) 341 - 9959 - 750

Anschrift:
Inselstr. 22
04103 Leipzig

Sekretariat:
Email, Telefon/Fax:
+49 (0) 341 - 9959
- 752
- 658

# Summer 2022

## Representation theory of finite groups

• Lecturer: Arthur Bik
• Date: Thursdays 7:30-9:00 and Fridays 9:15-10:45
• Room: SG 3-10
• Keywords: Representation theory, Group actions, Characters
• Prerequisites: Basic knowledge about groups and vector spaces
• Remarks: https://personal-homepages.mis.mpg.de/arbik/reptheory22.html

## Abstract

Representation theory is about understanding and exploiting symmetry using linear algebra. The central objects of study are linear actions of groups on vector spaces. This gives rise to a very structured and beautiful theory. The aim of this course dealing with finite groups and complex vector spaces is to introduce this theory. Representation theory plays a major role in mathematics and physics. For example, it provides a framework for understanding finite groups, special functions, and Lie groups and algebras. In number theory, Galois groups are studied via their representations; this is closely related to modular forms. In physics, representation theory is the mathematical basis for the theory of elementary particles. After introducing the concept of a representation of a group, we will study decompositions of representations into irreducible constituents. A finite group only has finitely many distinct irreducible representations; these are encoded in a matrix called the character table of the group. One of the goals of this course is to use representation theory to prove Burnside's theorem on solvability of groups whose order is divisible by at most two prime numbers. Another goal is to construct all irreducible representations of the symmetric group.

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## First-order and online optimization methods

• Lecturer: Katerina Papagiannouli, André Uschmajew
• Date: Lectures: Tuesdays 11:00-12:30, Exercises (biweekly): Tuesdays 14:00-15:00
• Room: MPI MiS G3 10
• Keywords: online convex optimization, optimization on manifolds, multi-armed bandit, games and saddle point problems
• Prerequisites: Basics of linear algebra, analysis, and probability
• Remarks: The class will start on 19/04; 9-12 lectures

## Abstract

In this class we will study first-order optimization methods for constrained and unconstrained optimization methods. In addition, a major part of the lecture will be devoted to aspects of online convex optimization, which is a combination of convex optimization, statistical learning, and game theory. Online optimization is motivated from practical applications in which the environment is so complex that it is difficult to design robust optimization models and apply classic algorithmic theory. In the online optimization framework, the optimization is instead considered as a process that learns from experience as one goes along and more aspects of the problem are observed. In the exercise class (on demand) a practical application to recommender systems will be considered.

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## IMPRS Ringvorlesung

08.08.2022, 02:30