2024/2025
Research Seminar "Modern Dynamical Systems"
Type:
Optional course (faculty)
When:
3, 4 module
Open to:
students of all HSE University campuses
Language:
English
Course Syllabus
Abstract
Dynamical systems in our course will be presented mainly not as an independent branch of mathematics but as a very powerful tool that can be applied in geometry, topology, probability, analysis, number theory and physics. We consciously decided to sacrifice some classical chapters of ergodic theory and to introduce the most important dynamical notions and ideas in the geometric and topological context already intuitively familiar to our audience. As a compensation, we will show applications of dynamics to important problems in other mathematical disciplines. We hope to arrive at the end of the course to the most recent advances in dynamics and geometry and to present (at least informally) some of results of A. Avila, A. Eskin, M. Kontsevich, M. Mirzakhani, G. Margulis.In accordance with this strategy, the course comprises several blocks closely related to each other. The first three of them (including very short introduction) are mainly mandatory. The decision, which of the topics listed below these three blocks would depend on the background and interests of the audience.http://crei.skoltech.ru/app/data/uploads/sites/42/2021/06/MA060257_skripchenko.pdf
Learning Objectives
- Students will read and understand deeply plenty of celebrated papers in dynamics. Some of the materials we are supposed to study were mentioned by the Fields committee as a main motivation to award the Fields medal in the last few decades.
Expected Learning Outcomes
- A student learns how to use basic tools of hyperbolic geometry, can classify the isometries of the hyperbolic plane, is able to provide some explicit examples of Fuchsian groups etc.
- The student shows ability to use key concepts of the course
- The student shows ability to use key concepts of the course in real mathematical research
- Students are supposed to become familiar with the most advanced techniques that are applied in dynamical systems.
Course Contents
- Introduction
- Dynamics and geometry
- Dynamics and topology
- Dynamics and number theory
- Dynamics and analysis.
- Dynamics and probability.
