Александр Моисеевич Вербовецкий, Иосиф Семенович Красильщик
Когомологические аспекты геометрии дифференциальных уравнений
Семинары проходят по средам, в 19:20 либо очно в аудитории 303, либо в зуум.
Плейлист семинара - на YouTube и на RuTube
ВЕСНА 2026
18 марта 2026 (среда), 19:20, семинар пройдёт только в Zoom'е:
Meeting ID: 88 17 12 1842
Passcode можно узнать по почте seminar@gdeq.org
Докладчик: Raffaele Vitolo
Тема: Bi-Hamiltonian systems from homogeneous operators
Язык доклада: английский
Аннотация:
Many "famous" integrable systems (KdV, AKNS, dispersive water waves, etc.) have a bi-Hamiltonian pair of the following form: A_1 = P_1 + R_k and A_2 = P_2, where P_1, P_2 are homogeneous first-order Hamiltonian operators and R_k is a homogeneous Hamiltonian operator of degree (order) k. The Hamiltonian property of P_1, P_2 and their compatibility were given an explicit analytic form and geometric interpretation long ago (Dubrovin, Novikov, Ferapontov, Mokhov). The Hamiltonian property of R_k was studied in the past (Doyle, Potemin; k=2,3) and recently revisited with interesting results.
In this talk, we illustrate the analytic form and some preliminary geometric interpretation of the compatibility conditions between P_i and R_k, k=2,3.
See the recent papers
https://arxiv.org/abs/2602.14739
https://arxiv.org/abs/2407.17189
https://arxiv.org/abs/2311.13932
Joint work with P. Lorenzoni and S. Opanasenko.
25 февраля 2026 (среда), 19:20, семинар пройдёт очно в Независимом университете, комн. 303, начало в 19:20, одновременно будет трансляция в Zoom'е:
Meeting ID: 88 17 12 1842
Passcode можно узнать по почте seminar@gdeq.org
Докладчик: А.А. Дуюнова
Тема: Differential invariants and quotient of the Euler equations on a sphere
Язык доклада: английский
Аннотация:
We consider the Euler system on a sphere written in stereographic coordinates. Since the system is underdetermined we consider flow of a medium taking into account thermodynamic equations of state.
Lie algebras of symmetries of the Euler system are found and we give their classification depending on possible equations of state. Among these Lie algebras there is one that preserves any thermodynamic equation. Such symmetries and the corresponding rational differential invariants we call kinematic. The field of kinematic differential invariants is described: basis differential invariants as well as invariant derivations are found. Then we find relations (syzygies) between the second-order invariants, from which we find a quotient equation for the Euler system on a sphere.
18 февраля 2026 (среда), 19:20, семинар пройдёт очно в Независимом университете, комн. 303, начало в 19:20, одновременно будет трансляция в Zoom'е:
Meeting ID: 88 17 12 1842
Passcode можно узнать по почте seminar@gdeq.org
Докладчик: В.Н. Рубцов
Тема: 2-Valued algebraic groups, the Chazy equation, and quasimodular forms
Язык доклада: английский
Аннотация:
I will discuss some (un)known relations between the objects in the title.
In particular, the celebrated Chazy equation emerges as an associativity condition.
The talk is based on ongoing joint work with V. Buchstaber and M. Kornev (Steklov Mathematical Institute, RAS).
11 февраля 2026 (среда), 19:20, семинар пройдёт очно в Независимом университете, комн. 303, начало в 19:20, одновременно будет трансляция в Zoom'е:
Meeting ID: 88 17 12 1842
Passcode можно узнать по почте seminar@gdeq.org
Докладчик: В.В. Лычагин
Тема: On the management of thermodynamic processes
Язык доклада: английский
Аннотация:
At the beginning of the talk, three geometric approaches to thermodynamics will be discussed.
The first approach is the Gibbs energy approach, which will be reformulated in terms of contact geometry and where the description of substances (the so-called equations of state) is given in terms of Legendre submanifolds in thermodynamic contact phase spaces, and thermodynamic processes, as well as controls, will be given by contact vector fields.
The second approach is based on information geometry and follows the principle of maximum entropy, also known as the principle of minimum information gain or Occam's razor. Both of these approaches lead us to the same model of thermodynamics, but they also introduce important new concepts, such as the Gibbs-Duhem principle and Riemannian structures on Legendre submanifolds.
The third approach is based on the geometry of jet spaces (or the geometry of differential equations), and it provides a more convenient apparatus for the practical description and calculation of both equations of state and thermodynamic processes, taking into account phase transitions.
Thermodynamic process control will be understood as a thermodynamic process that does not destroy the process in question, but allows it to be accelerated or slowed down.
The set of controls forms a Lie algebra, in which the Lie algebra of symmetries is a Lie subalgebra.
We will present equations that depend on the equations of state of the medium and allow us to find control processes, as well as illustrate their application in the case of adiabatic processes.
If time permits, phase transitions of the first, second and higher orders will be considered, both in thermodynamic processes and in controls, as well as their connection with Arnold's theory on the singularities of projections of Lagrangian manifolds.