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Обычная версия сайта
2019/2020

Научно-исследовательский семинар "Представления и вероятность 1"

Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус: Дисциплина общефакультетского пула
Когда читается: 1, 2 модуль
Язык: английский
Кредиты: 3
Контактные часы: 30

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

Learning Objectives

  • Knowledge of key notions and results in ergodic theory for group actions.
  • Knowledge of key notions and results in theory of stochastic differential equations and related systems.
Expected Learning Outcomes

Expected Learning Outcomes

  • Knowledge of key results in theory of stochastic processes. Ability to apply them to study various processes, and to construct processes with desired properties.
  • Knowledge of main methods and results in theory of SDEs. Ability to solve and analyze simple SDEs, including those arising in other brances of mathematics.
  • Knowledge of basic results in ergodic theory of group actions for various classes of groups (amenable, hyperbolic, etc.), and its relations with the theory of Markov operators.
Course Contents

Course Contents

  • Elements of theory of stochastic processes
    Main notions in the theory of random processes (independence of differences, covariation function, trajectory-wise properties of processes). Key examples: Poisson flow, Brownian motion, and thier properties.
  • Elements of theory of stochastic differential equations
    Stochastic DEs. Ito and Stratonovitch integrals. Emergence of SDE in problems of asymptotic representations theory.
  • Elements of ergodic theory of group actions.
    Examples of group actions with invariant measure. Group properties and behavior of averages. Amenability. Ergodic theorems for various classes of groups.
Assessment Elements

Assessment Elements

  • non-blocking Activities during classes
  • non-blocking Exam
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.4 * Activities during classes + 0.6 * Exam
Bibliography

Bibliography

Recommended Core Bibliography

  • Dobrow, R. P. (2016). Introduction to Stochastic Processes with R. Hoboken, New Jersey: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1214883

Recommended Additional Bibliography

  • Hasselblatt, Boris. Ergodic Theory and Negative Curvature [Электронный ресурс] / Boris Hasselblatt; БД springer. - Springer, Cham, 2017 - ISBN: 978-3-319-43058-4 (Print).
  • Nielsen, S. R. K., & Zhang, Z. (2017). Stochastic Dynamics. Aarhus, Denmark: Aarhus University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1809724