Course detail
Invariants and Symmetry
FSI-9ISYAcad. year: 2024/2025
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
Mode of study
Guarantor
Department
Entry knowledge
Rules for evaluation and completion of the course
Lectures: recommended
Aims
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.
Study aids
Prerequisites and corequisites
Basic literature
Healey, Richard. Gauging what's real: The conceptual foundations of contemporary gauge theories. Oxford University Press on Demand, 2007
Mansfield, E. L., A practical guide to the invariant calculus. Cambridge University Press, 2010 (EN)
Olver, P. J., Equivalence, invariants and symmetry. Cambridge University Press, 1995 (EN)
Recommended reading
Classification of course in study plans
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
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