Course detail

Invariants and Symmetry

FSI-9ISYAcad. year: 2020/2021

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.

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.


Knowledge of linear algebra and algebra, especially vector spaces and group theory.


Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Olver, P. J., Equivalence, invariants and symmetry. Cambridge University Press, 1995 (EN)
Mansfield, E. L., A practical guide to the invariant calculus. Cambridge University Press, 2010 (EN)
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 (EN)
Healey, Richard. Gauging what's real: The conceptual foundations of contemporary gauge theories. Oxford University Press on Demand, 2007

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.

Language of instruction

Czech, English

Work placements

Not applicable.


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

Classification of course in study plans

  • Programme D-APM-K Doctoral, 1. year of study, summer semester, 0 credits, recommended

  • Programme D4P-P Doctoral

    branch D-APM , 1. year of study, summer semester, 0 credits, recommended

Type of course unit



20 hours, optionally

Teacher / Lecturer


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