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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

Opinion survey results Examination dates SS schedule Noticeboard

Annotation -

Last update: prof. RNDr. Vladimír Souček, DrSc. (13.09.2013)

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).

Course completion requirements -

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

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

Literature -

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

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

Teaching methods -

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

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

Requirements to the exam -

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

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

Syllabus -

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

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.