Speaker:  A. Alexandrov (IBS, Korea)

Date and time:  17.07.2024, 17:00 (GMT +03:00)

Title:  KP integrability in TR through the x-y swap relation

Abstract:  I will discuss a universal relation sometimes called the x-y swap relation, which plays a prominent role in the theory of topological recursion (TR). In particular, the $x-y$ swap relation is very natural for the KP integrability and can be described by certain integral transforms, leading to the Kontsevich-like matrix models. This allows us to establish general KP integrability properties of the TR differentials. This talk is based on a joint work with Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, and Sergey Shadrin.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker:  Sergei Igonin (Yaroslavl State University) 

Date and time:  19.06.2024, 17:00 (GMT +03:00)

Title: Discrete Miura-type transformations, gauge simplifications, and group actions associated with Lax pairs for differential-difference equations

Abstract:  In this talk I will discuss some relations between matrix differential-difference Lax pairs, gauge transformations, and discrete Miura-type transformations for differential-difference (lattice) equations.

I will present sufficient conditions for the possibility to simplify a matrix differential-difference Lax pair by local matrix gauge transformations.

Also, I will present a method to construct Miura-type transformations for differential-difference equations, using gauge transformations and invariants of Lie group actions on manifolds associated with Lax pairs of such equations.

The method is applicable to a wide class of Lax pairs.

The considered examples include the (modified) Volterra, Itoh-Narita-Bogoyavlensky, Toda lattice equations, a differential-difference discretization of the Sawada-Kotera equation, and Adler-Postnikov equations from [1]. Applying the method to these examples, one obtains new integrable nonlinear differential-difference equations connected with these equations by new Miura-type transformations.

Some steps of our method generalize (in the differential-difference setting) a result of V.G. Drinfeld and V.V. Sokolov [3] on Miura-type transformations for the partial differential KdV equation.

This talk is based on the preprint [4] and a joint paper with G. Berkeley [2].

[1] V.E. Adler, V.V. Postnikov. Differential-difference equations associated with the fractional Lax operators. J. Phys. A: Math. Theor. (2011) 44, 415203.

[2] G. Berkeley, S. Igonin. Miura-type transformations for lattice equations and Lie group actions associated with Darboux-Lax representations. J. Phys. A: Math. Theor. (2016) 49, 275201.  https://arxiv.org/abs/1512.09123

[3] V.G. Drinfeld, V.V. Sokolov. On equations that are related to the Korteweg-de Vries equation. Soviet Math. Dokl. (1985) 32, 361-365.

[4] S. Igonin. Simplifications of Lax pairs for differential-difference equations by gauge transformations and (doubly) modified integrable equations. (2024)  https://arxiv.org/abs/2403.12022

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker:  Alexander Mikhailov (Leeds University)

Date and time:  5.06.2024, 17:00 (GMT +03:00)

Title: Deformations of noncommutative algebras and non-Abelian Hamiltonian systems

Abstract: By a well-known procedure, usually referred to as ``taking the classical limit'', quantum systems become classical systems, equipped with a Hamiltonian structure (symplectic or Poisson). From the deformation quantisation theory we know that a formal deformation of a commutative algebra A leads to a Poisson bracket on A and that the classical limit of a derivation on the deformation leads to a  Hamiltonian derivation on A defined by the Poisson bracket. In this talk I present a generalisation of it for formal deformations of an arbitrary noncommutative algebra A [1]. A deformation leads in this case to a commutative Poisson algebra structure on П(A):=Z(A)\times(A/Z(A)) and to the structure of a $П(A)-Poisson module on A, where Z(A) denotes the centre of A. The limiting derivations are then still derivations of A, but with the Hamiltonians belong to П(A), rather than to A. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the  Kontsevich integrable map, the quantum plane the quantised Grassmann algebra and quantisations of the  Volterra hierarchy [2], [3], [4].

This talk is based on a joint work with Pol Vanhaecke [1].

[1] Alexander V. Mikhailov and Pol Vanhaecke.  Commutative Poisson algebras from deformations of noncommutative algebras. arXiv:2402.16191v2, 2024.

[2] Alexander V. Mikhailov, Quantisation ideals of nonabelian integrable systems. Russ. Math. Surv., 75(5):199, 2020.

[3] Sylvain Carpentier, Alexander V. Mikhailov and Jing Ping Wang. Quantisation of the Volterra hierarchy. Lett. Math. Phys., 112:94, 2022.

[4] Sylvain Carpentier, Alexander V. Mikhailov and Jing Ping Wang. Hamiltonians for the quantised Volterra hierarchy. arXiv:2312.12077, 2023.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker:  Veronica Fantini (IHES)

Date and time:  22.05.2024, 17:00 (GMT +03:00)

Title: An introduction to resurgence and ODEs with irregular singularities

Abstract: The fascinating theory of resurgence introduced by Écalle, studies divergent power series in the complex domain. Among its different applications, it has been considered in the study of ODEs with irregular singularities. In this talk I will discuss some of the main aspects of resurgence theory, focusing in particular on the ones concerning formal solutions of irregular singular ODEs.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker:  Georgy Sharygin (Lomonosov MSU)

Date and time:  8.05.2024, 17:00 (GMT +03:00)

Title: Noncommutative discrete integrable systems and recurrencies

Abstract: In the theory of integrable systems it is known that in many cases there are reductions that relate different systems of differential and difference equations; these reductions relate equations of the systems, send the Lax pairs of the systems to each other etc. It turns out that very much similar relations show up in the theory of noncommutative equations, where the algebra of (differentiable) functions is replaced by a noncomutative associative algebra endowed with a derivative (for instance the algebra of matrix-valued functions on a straight line) and discrete functions also take values in the same algebra. The examples include 2-dimensional discrete Toda system, Somos recurrencies, discrete Painleve equations and others. In my talk I will explain the main ideas behind these constructions. Based on a joint work with Irina Bobrova, Vladimir Rubtsov and Vladimir Retakh, arXiv:2311.11124v2.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker:  Eugene Rogozinnikov (Max-Planck-Institut für Mathematik in den Wissenschaften (MPI MiS), Leipzig, Germany)

Date and time:  10.04.2024, 17:00 (GMT +03:00)

Title: Hermitian Lie groups as symplectic groups over noncommutative algebras

Abstract: In my talk (based on a joint work with D. Alessandrini, A. Berenstein, V. Retakh and A. Wienhard), I introduce the symplectic group $Sp_2(A,\sigma)$ over a noncommutative algebra $A$ with an anti-involution $\sigma$ and show that many Lie groups can be seen in this way. Of particular interest will be the classical Hermitian Lie groups such as $Sp(2n,R)$, $U(n,n)$ and their complexifications. For these groups, I realize their symmetric space in terms of $Sp_2(A,\sigma)$ thus generalizing several famous models of the hyperbolic plane and the three-dimensional hyperbolic space. Our construction has a flavor of noncommutative projective line over the complexification of $A$ which is always a compact symmetric space when $A$ Hermitian and semisimple or its complexification. We expect it to hold for any semisimple $A$. This, in turn, would imply that $Sp_2(A,\sigma)$ is reductive when $A$ is semisimple.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker:  Ismagil Habibullin (Ufa Federal Research Centre of Russian Academy of Science)

Date and time:  27.03.2024, 17:00 (GMT +03:00)

Title: On the classification of nonlinear integrable chains in 3D

Abstract: Nonlinear integrable equations with three independent variables, at least one of which is discrete, may admit boundary conditions that include discontinuities in the discrete variable while still maintaining the integrability of the equation. This means that by imposing these boundary conditions on the two ends of a segment, one can obtain an integrable system of two-variable equations. Integrable systems have a broad range of boundary conditions that allow for discontinuities and still maintain the integrability properties of the system. In some cases, these discontinuity conditions result in systems that are soliton-like, and it has been shown that there exists a single special discontinuity condition that yields an integrable finite system in the Darboux sense. In our recent work, we have demonstrated that this type of special reduction, which can be of arbitrary order, can be successfully applied to solve the problem of classifying three-dimensional integrable systems.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker:  Vladimir Retakh (Rutgers University)

Date and time:  13.03.2024, 17:00 (GMT +03:00)

Title:  The noncommutative Laurent phenomenon

Abstract: The composition of polynomials is always a polynomial, but this is not the case for Laurent polynomials containing inverse variables. When Laurent property persists, one talks about the Laurent phenomenon. In the commutative case, the Laurent phenomenon is closely related to the theory of cluster algebras. I will talk about examples of the noncommutative Laurent phenomenon and its connection with the theory of noncommutative cluster algebras. The report is based on joint work with Arkady Berenstein.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker:  Vincent Caudrelier (School of Mathematics, University of Leeds)

Date and time:  28.02.2024, 17:00 (GMT +03:00)

Title: Soliton interactions, Yang-Baxter and reflection maps and their Poisson properties

Abstract: Using the vector nonlinear Schrödinger equation as the main example, I will briefly review how certain solutions of the set-theoretical Yang-Baxter equation, called Yang-Baxter maps, arise from the interactions of multicomponent solitons. This is best seen using the Zakharov-Shabat dressing method and refactorisation of the elementary dressing factors. In this purely classical context, it is remarkable that the Yang-Baxter equation also ensures that the total scattering map describing the collisions consistently factorises into a product of two-soliton collisions, just like in the more well-known quantum context. I will then discuss the problem of integrable boundary conditions and explain how it leads to the introduction of the set-theoretical reflection equation. Solutions to this equation, called reflection maps, arise from the reflection of a soliton on the boundary. Again, the complete analogy between this context and the more well-known quantum reflection equation introduced by Cherednik and Sklyanin holds. Finally, I will present results on the symplectic and Poisson properties of these maps. This is a natural problem to consider given the interpretation (reviewed e.g. in Faddeev-Takhtajan's book) of soliton dynamics in the scalar case as canonical transformations.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com

Speaker:  Alexandra Garazha (MSU, Moscow)

Date and time:  31.01.2024, 17:00 (GMT +03:00)

Title: Kronecker's method and complete systems of functions in bi-involution on classical Lie algebras

Abstract: We examine the relation of the integrability property and the bi-Hamiltonian structure, the latter means that there are two compatible Poisson brackets and the dynamics is Hamiltonian with respect to both of them. We study the simplest case when one bracket is linear and the other one is constant. Namely, we will consider a classical simple Lie algebra g and define two Poisson brackets: the classical Lie-Poisson bracket and the bracket "with frozen argument A", which can be constructed for any A ∈ g. 

If A is regular, the corresponding complete systems of functions in bi-involution can be obtained by the Mishchenko-Fomenko argument shift method. We will show how to generalize this method to the case of an arbitrary element A using an algebraic approach.

To access the online seminar please contact  Anna Tolbey bekvaanna@gmail.com