Probability Theory
with Stanislav Apostolov and Andrei Sobolevski
HSE University, Faculty of Physics (2024-2026)
The course is aimed at developing students' skills in working with random variables and stochastic processes. The presentation of the formal theoretical framework is accompanied by intuitive explanations of key constructions and results. In defining the fundamental concepts – such as random variables, probability, and probability density (distribution function) – an intuitive approach is adopted, without recourse to measure theory. A distinctive feature of the course is the extensive use of concrete examples of probability distributions to illustrate general theoretical principles. The central part of the course focuses on asymptotic results in probability theory, including the law of large numbers, the classical central limit theorem and its generalization to stable laws, as well as limit theorems for extreme value distributions and large deviations. The final section covers standard topics in the fundamentals of Markov chain theory. The study of this discipline is based on prior knowledge acquired in courses on Mathematical Analysis, Linear Algebra, Differential Equations, and Elements of the Mathematical Methods of Physics.
Stochastic Processes and Modeling in Physics
with Sergey Belan and Michael Chertkov
Skoltech (2016-2017)
HSE University, Faculty of Physics (2021-2023)
The course offers as a soft and self-contained introduction to modern applied probability covering theory and application of stochastic models. Emphasis is placed on intuitive explanations of the theoretical concepts such as random walks, the law of large numbers, Markov processes, mutual information, Shannon's entropy, etc., supplemented by computational implementations of basic algorithms. Most of the discussed concepts are illustrated with examples from natural sciences. To successfully master the discipline, students must have basic skills in creating programs in any programming language.
Basics of Functional Integration
with Sergey Vergeles
HSE University, Faculty of Physics (2021-2023)
The purpose of this course is to master basic knowledge of path integrals in quantum mechanics and statistical physics. The course begins with defining the notion of a path integral and demonstrating its correspondence to quantum mechanics using examples of free particle motion and harmonic oscillator. Then we develop the quasi-classical approximation and consider the problems of tunneling and level splitting in a double-well potential. After that, the functional integration technique is applied to classical systems, and here we discuss the Brownian motion and conformations of long polymer chains.
Lecture Notes in Russian: