Семинары проходят по средам, в 19:20 либо  очно в аудитории 303, либо в зуум.

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26 марта 2025 (среда), 19:20, полностью в Zoom'е:
Meeting ID: 88 17 12 1842
Passcode можно узнать по почте seminar@gdeq.org
Докладчик: К.П. Дружков
Тема: A non-trivial conservation law with a trivial characteristic

Язык доклада: английский

Аннотация:
As far as I am aware, no nontrivial conservation laws surviving to the second page of Vinogradov's C-spectral sequence have been established. It turns out that presymplectic structures that cannot be described in terms of cosymmetries produce such conservation laws for closely related overdetermined systems. In particular, the presymplectic structure D_x of the potential mKdV equation gives rise to such a conservation law for the overdetermined system u_t = 4u_x^3 + u_xxx, u_y = 0. While this example is somewhat degenerate, it may be one of the simplest systems exhibiting this phenomenon. References: https://arxiv.org/abs/2502.11502

19 марта 2025 (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: Г.И. Шарыгин
Тема: Geometry of the full symmetric Toda system

Язык доклада: английский

Аннотация:
Full symmetric Toda system is the Lax-type system \dot L = [M(L),L], where the variable L is a real symmetric n x n matrix and M(L) = L+ - L- denotes its "naive" anti-symmetrisation, i.e., the matrix constructed by taking the difference of strictly upper- and lower-triangular parts L+ and L- of L. This system has lots of interesting properties: it is a Liouville-integrable Hamiltonian system (with rational first integrals), it is also super-integrable (in the sense of Nekhoroshev), its singular points and trajectories represent the Hasse diagram of Bruhat order on permutations group. Its generalizations to other semisimple real Lie algebras have similar properties. In my talk I will sketch the proof of some of these properties and will describe a construction of infinitesimal symmetries of the Toda system. It turns out that there are many such symmetries, their construction depends on representations of sl_n. As a byproduct we prove that the full symmetric Toda system is integrable in the sense of Lie-Bianchi criterion.

The talk is based on a series of papers joint with Yu.Chernyakov, D.Talalaev and A.Sorin.

12 марта 2025 (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: Turbulence geometry and Navier-Stokes equations

Язык доклада: английский

Аннотация:
It is proposed to consider turbulent media and, in particular, random vector fields from a geometric point of view. This leads to a geometry similar to, but not identical to, Finsler's.
We show that a turbulence generates a horizontal differential symmetric 2-form on the tangent bundle, which we call the Mahalanobis metric.
Thus, vector fields on the underlying manifold generate Riemannian structures on it by the restriction of the Mahalanobis metric on the graphs of vector fields.
In the case of so-called Gaussian turbulences, these Riemannian structures coincide and generate a unique Riemannian structure.
Moreover, similar to Finsler geometry, turbulence generates a nonlinear connection in the tangent bundle but the Mahalanobis metric generates an affine connection in the tangent bundle.
This affine connection and the Mahalanobis metric give us everything we need to construct the Navier-Stokes equations for turbulent media.
We will present two implementations of this scheme: for the flow of ideal gases and plasma, where turbulence is described by the Maxwell-Boltzmann distribution law, and compare them with the standard Navier-Stokes equations.

5 марта 2025 (среда), 19:20, полностью в Zoom'е:
Meeting ID: 88 17 12 1842
Passcode можно узнать по почте seminar@gdeq.org
Докладчик: К.П. Дружков
Тема: Invariant reduction for PDEs. II: The general mechanism

Язык доклада: английский

Аннотация:
en a local (point, contact, or higher) symmetry of a system of partial differential equations, one can consider the system that describes the invariant solutions (the invariant system). It seems natural to expect that the invariant system inherits symmetry-invariant geometric structures in a specific way. We propose a mechanism of reduction of symmetry-invariant geometric structures, which relates them to their counterparts on the respective invariant systems. This mechanism is homological and covers the stationary action principle and all terms of the first page of the Vinogradov C-spectral sequence. In particular, it applies to invariant conservation laws, presymplectic structures, and internal Lagrangians. A version of Noether's theorem naturally arises for systems that describe invariant solutions. Furthermore, we explore the relationship between the C-spectral sequences of a system of PDEs and systems that are satisfied by its symmetry-invariant solutions. Challenges associated with multi-reduction under non-commutative symmetry algebras are also clarified.

26 февраля 2025 (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: Д.А. Рудинский
Тема: Weak gauge PDEs

Язык доклада: английский

Аннотация:
Gauge PDEs are flexible graded geometrical objects that generalise AKSZ sigma models to the case of local gauge theories. However, aside from specific cases - such as PDEs of finite type or topological field theories - gauge PDEs are inherently infinite-dimensional. It turns out that these objects can be replaced by finite dimensional objects called weak gauge PDEs. Weak gauge PDEs are equipped with a vertical involutive distribution satisfying certain properties, and the nilpotency condition for the homological vector field is relaxed so that it holds modulo this distribution. Moreover, given a weak gauge PDE, it induces a standard jet-bundle BV formulation at the level of equations of motion. In other words, all the information about PDE and its corresponding BV formulation turns out to be encoded in the finite-dimensional graded geometrical object. Examples include scalar field theory and self-dual Yang-Mills theory.

19 февраля 2025 (среда), 19:20, полностью в Zoom'е:
Meeting ID: 88 17 12 1842
Passcode можно узнать по почте seminar@gdeq.org
Докладчик: Jacob Kryczka
Тема: Singularities and Bi-complexes for PDEs

Язык доклада: английский

Аннотация:
Many moduli spaces in geometry and physics, like those appearing in symplectic topology, quantum gauge field theory (e.g. in homological mirror symmetry and Donaldson-Thomas theory) are constructed as parametrizing spaces of solutions to non-linear partial differential equations modulo symmetries of the underlying theory. These spaces are often non-smooth and possess multi non-equidimensional components. Moreover, when they may be written as intersections of higher dimensional components they typically exhibit singularities due to non-transverse intersections. To account for symmetries and provide a suitable geometric model for non-transverse intersection loci, one should enhance our mathematical tools to include higher and derived stacks. Secondary Calculus, due to A. Vinogradov, is a formal replacement for the differential calculus on the typically infinite dimensional space of solutions to a non-linear partial differential equation and is centered around the study of the Variational Bi-complex of a system of equations. In my talk I will discuss a generalization in the setting of (relative) homotopical algebraic geometry for possibly singular PDEs.

This is based on a series of joint works with Artan Sheshmani and Shing-Tung Yau.