Description:

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.

Contents:

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. CohomologyofLiealgebras

Seminar contents:

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

Recommended literature:

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.

Keywords:

Algebraic topology, homotopy, homological algebra, cohomology, category theory

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar