**Department of Algebra and Geometry, Palacky University, Olomouc**

**PhD Studies in Geometry**

**Supervisor**: Prof. RNDr. Demeter Krupka, DrSc.

**Topics in Geometry and Global Analysis**

**Natural bundles, differential invariants, jets and contact elements**The aim is to investigate the geometric structure of higher order Grassmann bundles, the corresponding structure groups and differential invariants. Applications in the geometry of submanifolds and the calculus of variations can also be considered.

**Key words and phrases**: Jet groups (differential groups), differential invariants as equivariant mappings with respect to differential groups, the structure of manifolds of jets, higher order contact elements, higher order Grassmann bundles, invariants of submanifolds, applications in geometry and physics (natural variational principles).

**References (selection)**

- D.R. Grigore, D. Krupka, Invariants of velocities and higher order Grassmann bundles,
*J. Geom. Phys. 24 (1998), 244-264*. - I. Kolar, P. Michor, J. Slovak,
*Natural Operations in Differential Geometry*, Springer, Berlin, 1993. - D. Krupka, J. Janyska,
*Lectures on Differential Invariants*, Brno University, Brno, Czech Republic, 1990. - D. Krupka, M. Krupka, Jets and contact elements,
*Proc. Semin. on Diff. Geom., Mathematical Publications Vol. 2, Silesian University in Opava, Czech Republic, Opava, 2000, 39-85*. - A. Nijenhuis, Natural bundles and their general properties, in
*Differential Geometry, in honor of K. Yano, Kinokuniya, Tokyo, 1972, 317-334*. - T.Y. Thomas,
*The Differential Invariants of Generalized Spaces*, Cambridge, At the University Press, 1934.

- D.R. Grigore, D. Krupka, Invariants of velocities and higher order Grassmann bundles,
**Global variational principles in fibered spaces**The topic is devoted to open problems of the global variational theory in fibered spaces. The local and global inverse problem of the calculus of variations in field theory, as well as its generalizations will be considered. Research is based upon the variational sequence and the variational bicomplex theory.

**Key words and phrases**: Lagrange theory on fibered spaces, variational sequence and its generalizations, variational principles for partial differential equations, locally and globally variational equations, regularity and Hamilton theory, variational sequence and cohomology.

**References (selection)**

- I. Anderson, T. Duchamp, On the existence of global variational principles,
*Am. J. Math. 102 (1980), 781-867*. - G. Bredon,
*Sheaf Theory*, McGraw-Hill, New York, 1967. - D. Krupka, Lepagean forms in higher order variational theory, in:
*Modern Developments in Analytical Mechanics, Proc. IUTAM-ISIMM Sympos., Turin, June 1982; Academy of Sciences Turin, 1993, 197-238*. - D. Krupka, Variational sequences on finite order jet spaces, in:
*Differential Geometry and its Applications, Proc. Conf. Brno, 1989; World Scientific, Singapore, 1990, 236-254*. - O. Krupkova,
*The Geometry of Ordinary Variational Equations*, Lecture Notes in Math. 1678, Springer, Berlin, 1997. - D. Saunders,
*The Geometry of Jet Bundles*, London Mathematical Society Lecture Note Series 142, Cambridge University Press, New York, 1989. - R. Vitolo, Finite order Lagrangian bicomplexes,
*Math. Proc. Cambridge Phil. Soc. 125 (1999)*.

- I. Anderson, T. Duchamp, On the existence of global variational principles,
**Invariant variational functionals and conservation laws**The aim is to study current problems in the theory of invariant variational principles on fibered spaces. The research will be focused on the Noether's symmetries in field theory, and on energy-momentum tensors for differential equations.

**Key words and phrases**: Natural (covariant) variational principles, theorems of E. Noether, variational principles on principal fibre bundles, conservation laws, energy-momentum tensors.

**References (selection)**

- A. Fernandes, P. L. Garcia, and C. Rodrigo, Stress-energy-momentum tensors for natural constrained variational problems,
*J. Geom. Phys. 49 (2004), 1-20*. - M.J. Gotay, J.E. Marsden, Stress-energy-momentum tensor and the Belinfante-Rosenfeld formula,
*Contemporary Mathematics 132 (1992) 367-392*. - M. Kriele,
*Spacetime Foundations of General Relativity and Differential Geometry*, Springer, Berlin, 2001. - D. Krupka, A geometric theory of ordinary first order variational problems in fibered manifolds, II. Invariance,
*J. Math. Anal. Appl. 49 (1975) 469-476*. - D. Krupka, Variational principles for energy-momentum tensors,
*Rep. Math. Phys. 49 (2002) 259-268*. - E. Noether, Invariante Variationsprobleme,
*Gottingen Nachr. 1918, 235*. - J. Munoz-Masque, M. Eugenia Rosado Maria, Integrability of the field equations of invatiant variational problems on linear frame bundles,
*J. Geom. Phys, 49 (2004), 119-155*. - A. Trautman, Noether equations and conservation laws,
*Comm. Math. Phys. 6 (1967) 248-261*.

- A. Fernandes, P. L. Garcia, and C. Rodrigo, Stress-energy-momentum tensors for natural constrained variational problems,

**Obligatory courses**

*Topology*(D. Krupka, O. Krupkova) (point-set topology, algebraic topology, sheaves).*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**

*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).*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).*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.

- Last updated: September 11
^{th}2009