M7300 Global analysis

Faculty of Science
Autumn 2024
Extent and Intensity
4/2/2. 10 credit(s). Recommended Type of Completion: zk (examination). Other types of completion: k (colloquium).
In-person direct teaching
Teacher(s)
Mag. Katharina Neusser, Ph.D. (lecturer)
prof. RNDr. Jan Slovák, DrSc. (lecturer)
Guaranteed by
prof. RNDr. Jan Slovák, DrSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 12:00–13:50 MS1,01016, Tue 12:00–13:50 MS1,01016
  • Timetable of Seminar Groups:
M7300/01: Mon 16:00–17:50 MS1,01016, K. Neusser
Prerequisites
Elementary differential and integral calculus and linear algebra
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
The goals are to generalize differential and integral calculus from open subsets of R^n to submanifolds of R^n and also to abstract manifolds with and without boundary, to discuss various geometric structures on manifolds focusing on Riemannian metrics and symplectic structures, and to learn some basics about Lie groups.
Learning outcomes
After completion of the course a student should have a solid and comprehensive knowledge about analysis on manifolds, should know the basics of Riemannian and symplectic geometry, and should be familiar with Lie groups.
Syllabus
  • Differential and integral calculus on submanifolds of R^n and on (abstract) smooth manifolds with and without boundary: vector fields and their local flows, differential forms, distributions and the Frobenius Theorem, and Stokes Theorem and its consequences.
  • Lie groups
  • Geometric structures on manifolds
  • Riemannian manifolds (Levi-Civita connection, geodesics and Riemannian curvature,...)
  • Symplectic manifolds (Darboux Theorem, Langrangian submanifolds, Poisson bracket,...)
Literature
    recommended literature
  • LEE, John M. Introduction to smooth manifolds. 2nd ed. New York: Springer, 2013, xv, 708. ISBN 9781441999818. info
  • MICHOR, Peter W. Topics in differential geometry. Providence: American Mathematical Society, 2008, xi, 494. ISBN 9780821820032. info
  • 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 written exam
Language of instruction
English
Further Comments
Study Materials
The course is taught annually.
The course is also listed under the following terms Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2025.
  • Enrolment Statistics (Autumn 2024, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2024/M7300