2020/2021

Research Seminar "Representations and Probability 2"

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 3, 4 module

Instructors: Andrey V. Dymov, Alexey Vladimirovich Klimenko, Mauro Mariani, Grigori Olshanski

Language: English

ECTS credits: 3

Contact hours: 36

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

• Talk
• Exam

Interim Assessment

• Interim assessment (4 module)
  The students are given a list of problems to solve at home for 7-14 days before the exam. Number of points for each problem, as well as the formula to transform the sum of points into the usual 0..10 grade is given along with the problems. On the exam student presents their solutions orally and answer some questions on the adjacent theory. Points can be deduced for poor knowledge of theoretical basis of the solutions.

Bibliography

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

Recommended Additional Bibliography

• Fuad Aleskerov, & Andrey Subochev. (2013). Modeling optimal social choice: matrix-vector representation of various solution concepts based on majority rule. Journal of Global Optimization, (2), 737. https://doi.org/10.1007/s10898-012-9907-2