Syllabus
Hodge theory
Scalar product on forms, Hodge dual, coderivative, de Rham-Laplace and Beltrami-Laplace operators. Hodge decomposition, potential and copotential, harmonics, cohomology.
Topological methods
Cohomology a homology groups, homotopy, fundamental group, homotopy equivalence, homotopy operator, contraction, Poincare lemma.
Riemann geometry in terms of forms
Exterior calculus (overview). Maxwell theory. Othonormal frames, Cartan structure equations, Ricci coefficients. Bianchi identities. Calculation of the curvature, example - Vaidya metric.
Geometry of Lie groups and algebras
Lie groups, construction of Lie algebra, exponential mapping, Killing metric, structure constants. Bi-invariant metric, measure, covariant derivative. Adjoint representations. The action of Lie group on a manifold, flows and their generators. Representations on vector spaces.
Fibre bundles
Abstract fibre bundles. Vector bundles and their geometry, covariant derivative, vector potential and curvature. Objects on the gauge-algebra bundle.
Geometry of gauge fieldsInner degrees of freedom and their description in terms of vector bundles. Gauge symmetry. Gauge group and gauge algebra bundles. Gauge and Yang-Mills fields. The action and field equations. Electromagnetic and charged fields.
Characteristic classes
Invariant symmetric polynomials in curvature, Chern-Weil theorem, characteristic classes, Chern class and character, Pontrjagin class, Euler form, integral quantities.
Two-component spinorsSpace of spinors, antisymmetric metric, soldering form. Relation between spinors and vectors. Geometric quantities and physical fields in terms of spinors. Electromagnetic field and curvature.
Annotation
Riemann geometry in terms of forms, Hodge theory, topological methods. Lie groups and algebras.
Fibre bundles, geometry of gauge fields, characteristic classes. Two-component spinors.