MU:MUDGGA Geometry and Global Analysis - Course Information

MUDGGA Geometry and Global Analysis

Extent and Intensity

0/0. 0 credit(s). Type of Completion: -.

Guaranteed by

doc. RNDr. Michal Marvan, CSc.
Mathematical Institute in Opava

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

• Geometry and Global Analysis (programme MU, P1102) (2)
• Geometry and Global Analysis (programme MU, P1102) (2)

Course objectives

To verify whether the student has gained knowledge and skills needed for independent scientific work.

Syllabus

• 1. Fundamentals of the analysisi on manifolds:
  The algebra of smooth functions, vector and tensor fields, Lie bracket, integrable distribution. Exterior forms, integration on manifolds, Stokes theorem.
  The flow of a vector field, Lie derivative. Fundamentals of the theory of Lie groups and Lie algebras. De Rham cohomology.
  Fundamentals of Riemannian geometry. Jet spaces. Fundamentals of the calculus of variations.
  2. Theory of Lie groups and algebers:
  Lie groups and subgroups, Lie algebras, their ideals. Representations of Lie groups and algebras. G-modules and g-modules, their relationship. Nilpotent, solvable and semi-simple algebras. Fundamentals of the structure theory of simple algebras and their representations; weights and roots.
  Examples in complex and real domain, classical series.
  3. Homological algebra:
  Modules, chain complexes, exactness, resolvents and derived functors, Tor and Ext. Bicomplexes, spectral sequences. Homology and cohomology of diverse algebraic structures.
  4. Algebraic topology:
  The method of algebraic topology. Singular homology and cohomology, cell complexes and their (co)homology. Homotopy and homotopy groups, coverings, universal coverings. Generalized homology and cohomology theories, spectral sequences. Sheaves, abstract de Rham theorem.
  5. Riemannian geometry:
  Differential geometry of an immersed submanifold in Euclidean space, fundamental forms and basic equations. Manifolds with affine connection, geodesics, curvature and torsion tensor. Riemannian metric, metric connection, basic identities. Spaces of constant curvature. Gauss-Bonnet formula.
  6. Applications of differential geometry in mathematical physics:
  Geometric foundations of the general theory of relativity. Symplectic manifolds, Poisson manifolds, Hamiltonian formalism, Liouville theorem, action-angle variables. Calculus of variations, Euler-Lagrange equations, invariance and integrals of motion, Noether theorem.
  7. Geometric theory of differential equations:
  Jet spaces, Cartan distribution, formal integrability. Point, contact and higher symmetries, Lie algebra of symmetries. Conservation laws, horizontal cohomology. Coverings, nonlocal symmetries and conservation laws, Bäcklund transformations, zero curvature representations. Recursion operators, Hamiltonian structures, complete integrability.

Language of instruction

Czech

Further Comments

The course can also be completed outside the examination period.

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