Speaker: Yasushi Ikeda (Lomonosov Moscow State University)

Title: Quantum argument shifts in general linear Lie algebras

Date and time:  16.10.2024, 17:00 (GMT +03:00)

Abstract: Argument shift algebras in S(g) (where g is a Lie algebra) are Poisson commutative subalgebras (with respect to the Lie-Poisson bracket), generated by iterated argument shifts of Poisson central elements. Inspired by the quantum partial derivatives on U(gl_d) proposed by Gurevich, Pyatov, and Saponov, I and Georgy Sharygin showed that the quantum argument shift algebras are generated by iterated quantum argument shifts of central elements in U(gl_d). In this talk, I will introduce a formula for calculating iterated quantum argument shifts and generators of the quantum argument shift algebras up to the second order, recalling the main theorem.

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

Speaker:  Alexander Gaifullin (MIRAS, Moscow)

Date and time:  2.10.2024, 17:00 (GMT +03:00)

Title:  Involutive two-valued groups and their classification

Abstract: The theory of n-valued groups has been actively developed over the past decades, primarily in the works of V.M. Buchstaber, his co-authors and students. One of the interesting features of this theory is the very high complexity of classification problems. I will talk about a class of involutive two-valued groups for which classification problem have turned out to be available. The report will be based on joint work with V. M. Bukhstaber and A. P. Veselov.

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

Speaker:  Vladimir S. Gerdjikov (IMI, Bulgaria)

Date and time:  18.09.2024, 17:00 (GMT +03:00)

Title:  Riemann–Hilbert Problems and Integrable Equations

Abstract: The talk is based on the paper [1]. We provide an alternative approach to integrable nonlinear evolution equations producing new classes of such equations.

[1]  Gerdjikov, V.S.; Stefanov, A.A. Riemann-Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions. Symmetry 2023, 15, 1933. https://doi.org/10.3390/sym15101933

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

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