Course detail
Invariants and Symmetry
FSI-9ISY Acad. year: 2024/2025 Summer semester
The course is focused on the use of geometric methods in problems of differential equations and physics. The study of symmetries and equivalence problems requires a number of tools and techniques, many of which have their origins in differential geometry. Therefore, our study of differential equations and variational problems will have essentially a geometric character, unlike analytical methods. We will start with differential manifolds and Lie groups, the method of the moving frames will be essential here. We will focus on both the globally geometric view and also on calculations in local coordinates. Special attention will be paid to nonlinear problems. We will also study calibration invariants in connection with Maxwell's equations and quantum field theory.
Language of instruction
Czech
Supervisor
Department
Entry knowledge
Knowledge of linear algebra and algebra, especially vector spaces and group theory.
Rules for evaluation and completion of the course
The oral exam will test the knowledge of basic concepts and theorems and practical skills in solving geometric and physical problems.
Lectures: recommended
Aims
The aim is to master the differential geometry tools for solving invariance problems in applications.
The student will have an overview of the basic concepts and results of modern differential geometry. He will be able to use them in problems of solving differential equations, problems of variational calculus and physics.
The study programmes with the given course
Programme D-APM-K: Applied Mathematics, Doctoral, recommended course
Programme D-APM-P: Applied Mathematics, Doctoral, recommended course
Type of course unit
Lecture
20 hours, optionally
Syllabus
1. Smooth manifolds, vector fields
2. Distributions and foliations
3. Lie groups and Lie algebras
4. Representations
5. Jets and contact elements
6. Differential invariants
7. Symmetry of differential equations
8. Selected nonlinear problems
9. Classical and quantum field theory
10. Gauge invariants