Seminar for Geometry, education and visualization with applications
PROGRAM
MATEMATIČKI INSTITUT SANU
Seminar geometriju, obrazovanje i vizualizaciju sa primenama
PLAN RADA ZA APRIL 2022.
ČETVRTAK, 07.04.2022. u 17:15, On-line
V.Vedyushkina, Lomonosov Moscow State University
BILLIARD BOOKS AND THEIR CONNECTION WITH INVARIANTS OF INTEGRABLE SYSTEMS
Recently, the author has introduced a new class of integrable billiard books, which extends the well-known class of flat integrable billiards bounded by arcs of confocal quadrics.
"Billiard books" are glued from two-dimensional flat tables of integrable billiards (that is, bounded by confocal quadrics or concentric circles and their radii) along common arcs of the boundary. Permutations on the edges - the "spines" of the book - determine the transition of the ball from one sheet of the book to another after hitting the border of a flat piece. The design is well combined with the already known ones: a potential or a magnetic field is added.
The talk will focus on two subjects. First, these are the steps in proving A.T.Fomenko's conjecture on billiard modeling of integrable Hamiltonian systems. For an arbitrary non-degenerate singularity of an integrable system - a bifurcation of Liouville tori - we will show a billiard book that implements it. Such singularities-atoms correspond to the vertices of Fomenko-Zieschang molecules and define the bifurcation type of regular Liouville tori. In the second part of the talk, it is planned to highlight the topology of three-dimensional surfaces of constant energy for a number of systems of billiard books. The class of such manifolds turned out to be not limited to Seifert manifolds (as for integrable systems, Waldhausen manifolds were found, but not Seifert manifolds).
ČETVRTAK, 14.04.2022. u 17:15, On-line
Vladislav Kibkalo, Lomonosov Moscow State University
INTEGRABLE SYSTEMS OF DYNAMICS IN PSEUDO-EUCLIDEAN SPACE AND TOPOLOGY OF NON-COMPACT LIOUVILLE FOLIATIONS
Finite-dimensional integrable systems obtain a structure of Liouville foliation, i.e. their phase space is a disjoint union of common level surfaces (their connected components) of their integrals. Theory of topological classification of such systems and their singularities was built by A.Fomenko and his school. Integrable systems of rigid body dynamics, e.g. Euler, Lagrange and Kovalevskaya tops were analysed in this way and appeared to realize a lot of interesting effects possible in this theory.
Two properties of foliations important for the classification theorems are the compactness of the fibers (connected common levels of integrals) and completeness of Hamiltonian flows of their integrals (phase trajectory is well-defined for each real time). Extending the topological classification theory to other integrable systems is an open and fundamental problem.
As the first step it can be fruitful to obtain a set of examples of such foliations and analyze them. As it turns out, dynamical systems in a pseudo-Euclidean space (e.g. such analogs of famous integrable tops of Euler, Lagrange and Kovalevskaya) are also integrable and realize foliations with both compact and noncompact fibers and non-critical singularities (i.e. the foliation is not locally trivial but does not contain critical points of first integrals). We will discuss properties of such systems and topology of their Liouville foliations studied by the speaker.
ČETVRTAK, 28.04.2022. u 17:15, On-line
Djordje Kocic, Univerzitet u Beogradu, Matematicki fakultet
KLASIFIKACIJA POVRSI SA KONSTANTNIM UGLOM U ZAKRIVLJENOM PROIZVODU
Neka je $I\subset R$ otvoreni interval, f:I->R strogo pozitivna funkcija i E^2 Euklidska ravan. Klasifikovacemo sve povrsi zakrivljene proizvod mnogostrukosti I x_f E^2 za koje jedinicna
normala gradi konstantan ugao sa pravcem tangentnim na I.
Sednice seminara odrzavaju se u zgradi Matematickog instituta SANU, Knez Mihailova 36, na trecem spratu u sali 301f.
Rukovodilac Seminara dr Stana Nikcevic
