Introduction to Dynamical Systems and Statistical Mechanics: Perspectives from Ergodic Theory, Thermodynamic formalism, and Control Theory, 5 credits
Introduktion till dynamiska system och statistisk mekanik: Perspektiv från ergodteori, termodynamisk formalism och reglerteknik
Course information
Language of instruction: English
Course period: Spring 2026
Course structure: Hybrid: on-campus and digital participation
Recommended prerequisites
Linear Algebra, Real Analysis (or Geometry and Analysis III); solid familiarity with measure theory is recommended, though we will review key concepts at the start.
Learning outcomes
The course will provide doctoral students with foundational tools to understand the long-term behavior of dynamical systems using ergodic and statistical mechanics methods. After completing the course, students will be able to:
1) Understand core concepts of dynamical systems, ergodic theory, thermodynamic formalism, and their relevance for stability and control.
2) Understand statistical mechanics principles such as entropy, pressure, and Lyapunov exponents, and how they appear in both physical models and control theory.
3) Gain an understanding of ergodic optimization and variational principles in dynamical systems.
4) Recognize interdisciplinary applications in mathematics, physics, and engineering, and connect these concepts to their own research area through discussion and communication across disciplinary boundaries.
Learning outcomes for doctoral degree
This course is intended for doctoral students in Mathematics, Physics, and Information Technology, and will contribute to their doctoral degree by providing a shared theoretical foundation and interdisciplinary skills. After completing the course, students will:
- Develop a systematic understanding of ergodic and statistical mechanics approaches in dynamical systems.
- Gain the ability to analyze and synthesize dynamical models relevant to mathematics, physics, and control theory.
- Demonstrate the ability to communicate and discuss research-related concepts across disciplinary boundaries.
- Acquire insight into how ergodic and dynamical methods can be applied in interdisciplinary contexts and connected to their own research area.
Course contents
The course introduces dynamical systems, ergodic theory, and thermodynamic formalism with applications to statistical mechanics and control theory. Topics include:
- Basic measure-theoretic concepts and the weak* topology, providing the foundation for invariant measures and ergodic theory
- Long-term behavior of dynamical systems with motivating examples
- Invariant measures, ergodicity, entropy, Lyapunov exponents
- Thermodynamic formalism and variational principles
- Statistical mechanics models (e.g., Ising model) and their mathematical analysis
- Ergodic optimization and zero temperature
- Control theory applications, including the joint spectral radius, Lyapunov exponents, and connections to stability analysis
Instruction
The course is designed as an introductory, cross-disciplinary course accessible to all doctoral students, regardless of prior specialization. Lectures will provide clear introductions to central concepts in dynamical systems, control theory and statistical mechanics. To ensure accessibility, we begin with motivational examples and gradually progress to more formal treatments. Problem-solving sessions and interactive discussions will allow participants to deepen their understanding and prepare for the examination.
Assessment
In each session, a set of problems will be provided. Students are free to choose which problems to solve and must submit solutions to a total of 12 problems. Each problem is worth 10 points, and a total of 60 points is required to pass the course.
Extra points can be earned through active participation in the classes.
Course examiner
Reza Mohammadpour, reza.mohammadpour@math.uu.se
Department with main responsibility
Department of Mathematics
Contact person
Reza Mohammadpour, reza.mohammadpour@math.uu.se
Application
Submit the application for admission to: Reza Mohammadpour, reza.mohammadpour@math.uu.se
Submit the application not later than: