Algebraic Topology

Course guarantor:

Jan Vysoký

Lecturer:

Jan Vysoký

Tutor:

Jan Vysoký

Supervisor:

Department of Physics

Synopsis:

A study of modern mathematical and theoretical physics requires one to acquire an ever increasing knowledge of mathematical apparautus. The main goal of this course is to acquaint students with basic methods used in algebraic topology, namely elements of category theory, homototopies, homological algebra and cohomology. An important objective is to enhance the mathematical language by concepts appearing universally across disciplines like differential geometry and abstract algebra. During excercise sessions, students will try practical calculations of introduced mathematical structures.

Requirements:

The prerequisite (zápočet) is awarded for active participation in the exercises.

A student who wishes to take the exam must arrange a suitable date by email, at least one week in advance. An individual exam date will then be scheduled for the student by the instructor.

The exam proceeds as follows: The student is assigned three topics and given 60 minutes for written preparation. Afterwards, the student must demonstrate knowledge of these topics and may be further questioned by the examiner. The total duration of the exam is less than 3 hours.

Syllabus of lectures:

1. Homotopy relation

2. Fundamental group

3. Categories and functors

4. Cellular and simplicial komplexes

5. Simplicial and singular homology and their relation

6. de Rham cohomology,

7. Poincaré lemma and duality

8. Sheaves and associated Čech cohomology

9. Čech de Rham cohomology

10. Cohomology of Lie algebras

Syllabus of tutorials:

Practical calculations of introduced mathematical structures, proofs of simpler propositions.

Study Objective:

Knowledge:

Students will get acquainted with basic notions of algebraic topology and homological algebra.

Skills:

Students are able to understand applications of algebraic topology in theoretical physics.

Study materials:

Key references:

[1] L. W. Tu: Differential Geometry: Connections, Curvature, and Characteristic Classes. Vol. 275. Springer, 2017.

[2] R. Bott, L. W. Tu: Differential Forms in Algebraic Topology. Vol. 82. Springer Science & Business Media, 2013.

Recommended references:

[3] A. Hatcher: Algebraic Topology. Cambridge University Press, 2002.

[4] E. H. Spanier: Algebraic Topology. Vol. 55. No. 1. Springer Science & Business Media, 1989.

[5] E. Knapp, A. W. Knapp: Lie Groups, Lie Algebras, and Cohomology. Vol. 34. Princeton University Press, 1988.

Note:

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (elective course)
• Matematická fyzika (compulsory course in the program)