Research Group in Global Analysis and Applications

Department of Algebra and Geometry, Palacky University, Olomouc

PhD Studies in Geometry

Supervisor: Prof. RNDr. Demeter Krupka, DrSc.

Topics in Geometry and Global Analysis

Obligatory courses

  1. Topology (D. Krupka, O. Krupkova) (point-set topology, algebraic topology, sheaves).
  2. Differential geometry and Riemannian geometry (D. Krupka) (manifolds, vector bundles, calculus on manifolds, vector fields, Lie derivatives, Lie groups, principal and associated bundles, differential invariants, Riemannian geometry, connections).

Facultative courses

  1. Global Analysis (D. Krupka) (vector fields, differential forms, De Rham cohomology, Lie derivatives, fibred spaces, differential systems, differentiation and integration on smooth manifolds, jets and contact elements).
  2. Variational and geometric methods in physics (O. Krupkova) (variational principles in mechanics, higher order mechanics and field theory, Hamilton equations and their generalizations, canonical transformations and Hamilton-Jacobi theory, symmetries and conservation laws, constrained systems, geometric structure of the general relativity theory).
  3. The calculus of variations on smooth manifolds (D. Krupka) (jet prolongations of manifolds and fibered spaces, differential forms on jet spaces, global variational functionals, extremals, the local and global inverse problem of the calculus of variations, variationally trivial lagrangians, the variational sequence, invariant transformations and their generalizations, conservation laws and the Noether's theory, local and global variational principles).

Other duties

Seminar presentations, presentations at scientific meetings, journal publication of results.