SubjectsSubjects(version: 953)
Course, academic year 2023/2024
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Fibre Spaces and Gauge Fields - NMAG454
Title: Fibrované prostory a kalibrační pole
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:3/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: prof. Ing. Branislav Jurčo, CSc., DSc.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Geometry
Annotation -
The course is a continuation of the course 'Introduction to analysis on manifolds'. It is a basic course needed for a further study of differential geometry and global analysis, as well as for applications in mathematical physics (Yang-Mills fields).
Last update: Souček Vladimír, prof. RNDr., DrSc. (13.09.2013)
Course completion requirements -

Active participation in tutorials. Oral exam on the lecture material.

Last update: Jurčo Branislav, prof. Ing., CSc., DSc. (18.06.2021)
Literature -

L. Tu: Differential geometry, connections, curvature and characteristic classe, GTM 275, 2017

J. Lee: Introduction to smooth manifolds, Springer, GTM 218, 2013

R. W. Sharpe: Differential geometry. Cartan's generalization of Klein's Erlangen program, Springer, GTM 166, 1997

I. Kolář, P. Michor, J. Slovák: Natural operation in differential geometry, Springer, 2010

Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (25.01.2024)
Teaching methods -

Lectures combined with reading or presentations by particpating students that may be also on-line.

Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (25.01.2024)
Requirements to the exam -

Important topics will be subject of exam, which is oral with a written preparation or on-line eventually.

Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (25.01.2024)
Syllabus -

Smooth manifolds, differential forms. Distributions, Frobenius theorem (2 versions).

Fibre bundles, transition functions, vector fibre bundles, their local description, classifying maps.

Covariant derivatives on vector bundles, parallel transport, curvature, structure equations.

Homogeneous spaces, the Maurer--Cartan form, the Darboux derivative, fundamental theorem of calculus.

Principal fibre bundles, associated fiber bundles, principal connections and their curvature, structure equations.

Holonomy and monodromy. Calibration (Yang-Mills) fields.

Last update: Krýsl Svatopluk, doc. RNDr., Ph.D. (25.01.2024)
 
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