012s: Modern Geometrical and Topological Methods


Viktoria Vedyushkina, Lomonosov Moscow State University
Segey Zelik, HSE University & University of Surrey & Zhejiang Normal University
Vladislav Kibkalo, Lomonosov Moscow State University
Elena Nozdrinova, HSE University
Olga Pochinka, HSE University
Anna Tsvetkova, Ishlinsky Institute for Problems in Mechanics
Andrey Shafarevich, Lomonosov Moscow State University


14 May 2024 - 19 May 2024


The school is dedicated to the study of modern geometric and topological methods that are widely used in dynamical systems, mechanics and mathematical physics. The goal is to present to young researchers the latest results and achievements of leading experts in a detailed form, as well as to discuss several open problems of various aspects of dynamics.

On the one hand, we will discuss Hamiltonian and integrable systems, i.e. systems in which certain functions, such as energy, are preserved. We also will discuss how Hamiltonian systems can be applied to the semiclassical quantization technique and the construction of semiclassical asymptotics. If a system has an energy-independent conservation law, then Morse theory and topology of foliations are combined to describe the closures of trajectories of the system. Moreover, many well-known systems with this property have been modeled via generalized billiards in confocal quadrics.

Another direction of the school is associated with the case when a dynamical system may not have a first integral. Even in this case, it turns out to be possible to construct (under certain assumptions) an effective description of the system’s trajectories using invariants, including graph ones. When describing dynamics in a number of systems, we will also demonstrate and teach how to use special methods for modeling a system using programs. We will separately discuss how to describe bifurcations for various systems from the physics of our world.


All questions regarding participation in the school should be addressed to smc@sochisirius.ru