Invariants and Symmetry (FSI-9ISY)

Academic year 2022/2023
Supervisor: doc. RNDr. Miroslav Kureš, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech or English
Aims of the course unit:
The aim is to master the differential geometry tools for solving invariance problems in applications.
Learning outcomes and competences:
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.
Course contents:
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.
Teaching methods and criteria:
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.
Controlled participation in lessons:
Lectures: recommended
Type of course unit:
    Lecture  10 × 2 hrs. optionally                  
Course curriculum:
    Lecture 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
Literature - fundamental:
1. Olver, P. J., Equivalence, invariants and symmetry. Cambridge University Press, 1995
2. Mansfield, E. L., A practical guide to the invariant calculus. Cambridge University Press, 2010
3. Bocharov, A. V., Verbovetsky, A. M., Vinogradov, A. M., Symmetries and conservation laws for differential equations of mathematical physics. Providence, RI: American Mathematical Society, 1999
4. Healey, Richard. Gauging what's real: The conceptual foundations of contemporary gauge theories. Oxford University Press on Demand, 2007
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
D-APM-K combined study --- -- DrEx 0 Recommended course 3 1 S
D-APM-P full-time study --- -- DrEx 0 Recommended course 3 1 S