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Regular version of the site
2019/2020

## Research Seminar "Representations and Probability 2"

Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type: Optional course (faculty)
When: 3, 4 module
Instructors: Grigori Olshanski
Language: English
ECTS credits: 3

### Course Syllabus

#### Abstract

In recent decades several areas of mathematics were developed where constructions from the probability theory, the representations theory, or both play the central role. The seminar is focused on various topics in these domains, especially emphasizing connections between them.

#### Learning Objectives

• Knowledge of key notions and results in asymptotic representations theory
• Knowledge of key notions and results in theory of random point fields, including determinantal processes.

#### Expected Learning Outcomes

• Knowledge of key results in theory of determinantal processes. Ability to use them to study properties of simple DPs.
• Knowledge of key results and methods in asymptotic representations theory, including asymptotic theory of characters.
• Knowledge of main results in theory of othogonal polynomials (Christoffel-Darboux kernels, etc.) Familiarity with asymtotic results for orthogonal polynomial ensembles.

#### Course Contents

• Orthogonal polynomials and random point processes
Elements of theory of orthogonal polynomial ensembles. Their relations to random point processes. Asymptotic problems for point processes and the corresponding functional-analytic properties of the assocuated orthogonal polynomial ensembles.
• Determinantal processes
Definition. Correlation functions. Kernel of a process and the associated operator in L2. Macchi-Soshnikov theorem. Properties of DP: rigidity, behaviour of conditional measures.
• Asymptotic representations theory
Key problems in ART. Invariant measures on spaces of matrices. Spectres distributions and their asymptotics.

#### Assessment Elements

• Activities during classes
• Exam

#### Interim Assessment

• Interim assessment (4 module)
0.4 * Activities during classes + 0.6 * Exam

#### Recommended Core Bibliography

• Fulton, W. (1997). Young Tableaux : With Applications to Representation Theory and Geometry. Cambridge [England]: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=570403