Studium
Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev, prof. Krasil'shchik
Program v akademickém roce 2011/2012
24. 5. 2012
Mgr. Petr Vojčák, Recursion operators, nonlocal symmetries and related
structures for integrable systems (presentation of the Ph.D. thesis)
Prof. Iosif Krasil'shchik, DrSc., Open problems in integrable systems II
17. 5. 2012
Giovanni Moreno, Ph.D., Characteristics and singularities of solutions of nonlinear PDEs.
Abstrakt:
Prolongation procedure allows to frame nonlinear PDEs in the context of infinite jet bundles, where powerful homological methods can be exploited. However, by its definition, prolongation discards important information about singular solutions, which may have physical significance. In this talk I will show how to recover such information from the sequence of vertical bundles of an infinitely prolonged PDE.
Prof. Iosif Krasil'shchik, DrSc., Open problems in integrable systems I
10. 5. 2012
Maxim Pavlov, Ph.D. (Lebeděvův fyzikální ústav, Moskva, Rusko), Third local Lagrangian representation for the WDVV associativity equations with N=3.
Abstrakt:
For the first nonlocal (in negative direction) Hamiltonian operator of the system under study, we have found the associated Lagrangian representation of the latter.
3. 5. 2012
Prof. Iosif Krasil'shchik, DrSc., First integrals. Lie-Bianchi Theorem. Linearization and
computation of symmetries.
26. 4. 2012
Prof. Iosif Krasil'shchik, DrSc., Geometrization of ODEs. Geometry of distributions. Symmetries.
19. 4. 2012
Giovanni Moreno, Ph.D., Natural structures in iterated jet spaces.
Video
Abstrakt:
Mechanics is a well-known context where iterated (co)tangent bundles and their tautological structures play a prominent role. A similar role, in the geometrical theory of nonlinear PDEs, is played by iterated jet bundles. The very definition of higher-order jets can be seen as a natural equation in an iterated jet bundle. Moreover, it seems that such important notions as the equation of singularities of solutions and the space of initial data of a nonlinear PDEs, or the natural boundary conditions in variational calculus, can be given a natural geometric interpretation in the framework of iterated jet spaces.
10. 4. 2012
Maxim Pavlov, Ph.D. (Lebeděvův fyzikální ústav, Moskva, Rusko), Finite-component reductions of non-integrable kinetic equations.
Seminář se koná výjimečně v úterý od 14.45 hod. v místnosti LVT1. Všichni zájemci jsou srdečně zváni.
5. 4. 2012
Oleg Morozov, DrSc. (University of Tromsø, Norsko) Coverings, recursion operators, and nonlocal symmetries of integrable differential equations.
Video
Abstrakt:
The talk is devoted to recent results about recursion operators and nonlocal symmetries for nonlinear partial differential equations with more that two independent variables.
28. 3. 2012
Doc. RNDr. Zdeněk Dušek, Ph.D. (Univerzita Palackého v Olomouci) Existence of lightlike homogeneous geodesics.
Seminář se koná výjimečně ve středu od 14.45 hod. v místnosti R1. Všichni zájemci jsou srdečně zváni.
Abstrakt:
In previous projects, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point.
In the present project this affine method is refined and adapted to the pseudo-Riemannian and Lorentzian case. Results about existence of light-like homogeneous geodesics will be presented and illustrated with an example.
22. 3. 2012
RNDr. Jiřina Vodová, Low-order Hamiltonian operators having momentum.
15. 3. 2012
Giovanni Moreno, Ph.D., An introduction to jet spaces and their natural structures.
8. 3. 2012
Prof. Iosif Krasil'shchik, DrSc., Geometry of differential equations and integrable systems.
Video
Abstrakt:
I shall try to explain on elementary level how geometrical approach to PDEs helps to compute main structures responsible for integrability: recursion, Hamiltonian, and symplectic operators.
References:
Joseph Krasilshchik, Alexander Verbovetsky: Geometry of jet spaces and integrable systems, Geometry and Physics (2011),
doi:10.1016/j.geomphys.2010.10.012, http://arxiv.org/abs/1002.0077
1. 3. 2012
Doc. RNDr. Artur Sergyeyev, Ph.D., První seznámení se supersymetrií.
23. 2. 2012
Doc. RNDr. Michal Marvan, CSc., Operátory rekurze jako Bäcklundovy autotransformace linearizovaných rovnic.
1. 12. 2011
Doc. RNDr. Zdeněk Kočan, Ph.D., Diskrétní dynamické systémy a chaos.
24. 11. 2011
Dr. Blazej M. Szablikowski, (Univerzita Adama Mickiewicze, Poznan, Polsko), Classical r-matrix approach to Frobenius manifolds: taking advantage of Rota—Baxter and classical Yang—Baxter relations.
10. 11. 2011
Mgr. Jiří Jahn, RNDr. Jiřina Vodová, Singularity diferenciálních rovnic v komplexním oboru I.
3. 11. 2011
Doc. RNDr. Michal Marvan, CSc., Likvidace negramotnosti v Lobačevského geometrii.
20. 10. 2011
Doc. RNDr. Michal Marvan, CSc., Bourův problém II.
13. 10. 2011
Doc. RNDr. Michal Marvan, CSc., Bourův problém I.
6. 10. 2011
Doc. RNDr. Artur Sergyeyev, Ph.D., Několik poznámek o řešení nelineárních parciálních diferenciálních rovnic prvního řádu s jednou závisle proměnnou.
29. 9. 2011
Doc. RNDr. Artur Sergyeyev, Ph.D., Integrabilita Hamiltonovských dynamických systémů: od Hamiltona k Jacobimu přes Liouville'a.
22. 9. 2011
Mgr. Adam Hlaváč, Bäcklundova transformace pro rovnici konstantního astigmatismu.
15. 9. 2011
Dr. Sergey Zykov, (Universita di Salento, Italy), Classification of integrable hydrodynamic chains of triangular type.
Abstrakt:
Integrable quasilinear PDEs can be produced from hydrodynamic chains. In earlier work only a handful of examples of such chains was found.
In this talk we present the most general hydrodynamic chain which has the Riemann surface and therefore possesses an infinite number of conservation laws. The coefficients of the chain under study are parametrized using hypergeometric functions. The sequence of conservation laws is produced by a generating function.
Semináře se konají vždy ve čtvrtek od 14.45 hod. v budově Matematického ústavu, Na Rybníčku 1 v Opavě, v místnosti R1. Všichni zájemci jsou srdečně zváni.