## M7300 Global analysis

Faculty of Science
Autumn 2019
Extent and Intensity
4/2/2. 10 credit(s). Type of Completion: zk (examination).
Teacher(s)
Mag. Katharina Neusser, Ph.D. (lecturer)
Guaranteed by
prof. RNDr. Jan Slovák, DrSc.
Timetable
Mon 12:00–13:50 M6,01011, Thu 12:00–13:50 M3,01023
• Timetable of Seminar Groups:
M7300/01: Mon 16:00–17:50 M6,01011, R. Suchánek
Prerequisites
Elementary differential and integral calculus
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
The goals involve the geometric differential and integral calculus. Lie group and algebras will represent useful tools, including their representation. Selected applications will focus at concepts of Riemannian and symplectic geometries, optimal control, analytical mechanics and some further equations of mathematical physics.
Learning outcomes
The students will master the vector fields and exterior forms and the general Stokes theorem with applications in vector calculus. Furthe, they will actively learn basic elements of Lie theory involving the exponential mapping, adjoint representation, covering phenomena, and differential calculus for functions valued in Lie groups. They will be able to exploit the knowledge in analysis based on Riemannian and symplectic geometries, optimal control, and analytical mechanics.
Syllabus
• Vector fields and exterior forms on R^n a submanifolds of R^n,
• general Stokes theorem and its cosequences
• geometric theory of 1st oreder PDE (differencial ideals, Frobeniova theorem).
• Lie groups and subgroups, relations to Lie algebras (exponencial mapping, adjoint representation, coverings),
• Calculus for functions valued in Lie groups, basic concepts of representation theory, homogeneous spaces.
• Basics of Riemannian and symplecitc geometries.
• Applications in optimal control, analytical mechanics.
Literature
recommended literature
• AGRICOLA, Ilka and Thomas FRIEDRICH. Global analysis : differential forms in analysis, geometry and physics. Translated by Andreas Nestke. Providence, Rhode Island: American Mathematical Society, 2002. xiii, 343. ISBN 0821829513. info
• JOST, Jürgen. Riemannian geometry and geometric analysis. 5th ed. Berlin: Springer, 2008. xiii, 583. ISBN 9783540773405. info
Teaching methods
Standard lectures and seminars, independent study
Assessment methods
standard oral exam
Language of instruction
Czech