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
Кто читает:
Факультет математики
Где читается:
Факультет математики
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
- 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
- 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
- Elements of theory of stochastic processesMain 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 equationsStochastic 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.
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