Course detail
Invariants and Symmetry
FSI-9ISY Acad. year: 2022/2023 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
Learning outcomes of the course unit
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.
Prerequisites
Knowledge of linear algebra and algebra, especially vector spaces and group theory.
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline.
Assesment methods and criteria linked to learning outcomes
The oral exam will test the knowledge of basic concepts and theorems and practical skills in solving geometric and physical problems.
Aims
The aim is to master the differential geometry tools for solving invariance problems in applications.
Specification of controlled education, way of implementation and compensation for absences
Lectures: recommended
The study programmes with the given course
Programme D-APM-P: Applied Mathematics, Doctoral, recommended course
Programme D-APM-K: 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