NTMF060 - materials for the course in the spring term 2022

Materials for download:

Introductory remarks: video: [mp4 33MB 8min]
Hodge theory and de Rham cohomology: Local and global Hodge decomposition, harmonics. De Rham cohomology, cohomology groups, Betti numbers, Poincare duality. Examples of cohomology groups for the simplest manifols.; video: [mp4 484MB 115min]
Homology groups: Simplicial homology, simplicial complex, boundary operator, chains, cycles, boundaries. Homology groups, Betti numbers, Euler characteristic. Examples. Singular homology. Duality of homology and cohomology groups.; video: [mp4 719MB 165min]
Homotopy: Homotopy, homotopic invariants, homotopic type, deforamtion retract, contractibility using flow. Homotopic operator, Poincaré lemma. Map induced on homology groups, homotopy and homology. Fundamental group.; video: [mp4 491MB 109min]
Cartan calculus 1: Exterior calculus. Maxwell equations in forms. Cartan calculus. Basis of 1-forms, connection 1-forms, Ricci rotation coefficients, First Cartan structure equation. Curvature 2-forms, Riemann tensor, Second Cartan structure equation, Bianchi identities.; video: [mp4 207MB 122min]; notes: [pdf]
Cartan calculus 2: Gauss coordinates on a 2-sphere. Élie Cartan. Vaidya metric - the curvature tensor and field equations. Prahalad Chunnilal Vaidya. Censorship violation in radiation collapse.; video: [mp4 308MB 128min]; notes: [pdf]
Geometry of Lie groups 1: Lie groups, left and right invariant fields. Lie algebra of a Lie group, structure tensor, Killing metric, bi-invariant metric and measure. Exponential map. Adjoint representations. Simple and semisimple groups and algebras.; video: [mp4 536MB 107min]
Geometry of Lie groups 2: Left and right invariant covariant derivatives, Maurer-Cartan equations, $λ$ -derivative and Levi-Civita derivative, curvature. Action of Lie group on a manifold, generator of the action, Lie bracket. Representation of Lie group and algebra on a vector space, generator of representation. Group of isometries, Killing vectors and their transformations, example in $E^{2}$ and $L^{2}$ .; video: [mp4 485MB 112min]
[Example of the hyperbolic geometry has not been recorded.]; notes: Isometries of Euclidean and Hyperbolic geometry [pdf] (in Czech)
Vector bundles 1 - definition, covariant derivative: Fibre bundles, general definition, vector and tensor bundles, trivialization. Example of trivial and twisted bundles. Covariant derivative on vector bundles, pseudoderivative, derivative on tensor product of bundles. Covariant derivative for trivialization, vector potential. Vector-bundle valued antisymmetric forms, covariant exterior derivative.; video: [mp4 436MB 92min]; notes: [pdf] (in Czech)
Vector bundles 2 - curvature: Curvature operator, curvature tensor, the second covariant exterior derivative, curvature in terms of vector potential. Bianchi identity, local conservation law.; video: [mp4 286MB 67min]; notes: [pdf] (in Czech)
Gauge symmetry: Gauge symmetry, real vector bundle with O or SO symmetry. Metric structure on a fibre, Levi-Civita tensor, and orientation, orthonormal transformation, Lie group and algebra. Covariant derivative annihilating the metric, gauge transformation of the derivative, transformation of the vector potential and curvature tensor. Complex vector bundle and U symmetry. Antilinear operations and conjugated spaces. Scalar product, hermitian structure, unitary transformations. Covariant derivative consistent with the hermitian structure.; video: [mp4 419MB 106min]
Group and algebra bundles, associated bundles: Gauge group bundle, Gauge Lie algebra bundle. Covariant derivatives on the Lie algebra bundle. Adjoint representation of vector potentials and curvature tensors. Associated vector bundles. Representation of gauge group and algebra on vector bundles. Trivialization consistent with the association. Covariant derivative consistent with the association. Uniqueness of the derivatives indiced on the associated bundles. Information about the principal bundles.; video: [mp4 318MB 82min]
U(1) bundle and charged fields: One-dimensional complex vector bundle, canonical identifications in the tensor algebra, charged fields. U(1) gauge symmetry, operation with chred fields. Covariant derivative on charged fields, vector potential, curvature tensor. Gauge transformation of these objects.; video: [mp4 196MB 43min]; notes: [pdf]
Structure of the field theory: Description of the classical field theory, actions, equations of motions, stress-energy tensors.; video: [mp4 140MB 43min]
Chern-Weil theorem and characteristic classes: Invariant constant symmetric polynomials. Chern-Weil theorem. Characteristic classes, Chern class and character, Euler characteristic. Gauss-Bonnet theorem.; video: [mp4 508MB 112min]; notes: [pdf]
Spinors 1: Two component spinors, antisymmetric metric, soldering form, relation of spinors and tensors, EM field, Riemann tensor.; video: [mp4 206MB 131min]; notes: [pdf]
Spinors 2: Decomposition of a symmetric spinor, algebraic classification of the Weyl tensor, covariant derivative of spinors, Maxwell equations, geometrical interpretation of 1-spinors.; video: [mp4 181MB 105min]; notes: [pdf]

Geometrical Methods of Theoretical Physics II

materials for lectures

Information about lectures in spring term 2022:

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Lectures:

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