MUDGGA Geometry and Global Analysis

Mathematical Institute in Opava
Summer 2019
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
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.
The course is also listed under the following terms Winter 2009, Summer 2010, Winter 2010, Summer 2011, Winter 2011, Summer 2012, Winter 2012, Summer 2013, Winter 2013, Summer 2014, Winter 2014, Summer 2015, Winter 2015, Summer 2016, Winter 2016, Summer 2017, Winter 2017, Summer 2018, Winter 2018.
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