Speaker: V.E. Adler (L.D. Landau ITP)

Title: Around 3D-consistency

Date and time:  11.12.2024, 17:00 (GMT +03:00)

Abstract: The 3D-consistency property is usually formulated as the Consistency-Around-a-Cube for discrete equations on a square lattice (quad-equations). However, this property can be extended to some other types of equations, including continuous ones. In my talk, I will show that a multidimensional lattice governed by consistent quad-equations can carry some derivations that preserve this lattice and commute with each other. They are described by continuous equations of the KdV type and differential-difference equations of the Volterra lattice and dressing chain types, which are no less important objects than quad-equations. In principle, these equations can be obtained from quad-equations by continuous limits, but in my talk I will move in the opposite direction, interpreting quad-equations as the superposition formula for B\"acklund transformations. Particular attention will be paid to the interpretation of Volterra-type equations as negative symmetries for KdV-type equations and to the definition of 3D-consistency property for these symmetries.

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

Speaker:  Anastasia Doikou (Heriot-Watt University, Edinburgh, UK)

Title:  Set-theoretic Yang-Baxter equation, twists & quandle Hopf algebras

Date and time:  27.11.2024, 17:00 (GMT +03:00)

Abstract: The theory of the set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. The derivation of solutions of the braid equation via certain self-distributive structures called racks and quandles is reviewed. Generic, non-involutive set-theoretic solutions of the braid equation are then obtained from rack solutions by a suitable Drinfel'd twist, whereas all involutive solutions are obtained from the flip map via a twist. The universal algebras associated to both rack and generic set-theoretic solutions are also studied and the corresponding universal R-matrices are derived.

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

Speaker:  A. Kazakov (MSU, YarSU, KFU, MIPT)

Title:  Electrical networks from the Calderon problem to the phylogenetic networks

Date and time:  13.11.2024, 17:00 (GMT +03:00)

Abstract: An electrical network is essentially a graph with positive edge weights denoting conductivities. The graph nodes are divided into two sets: inner nodes and boundary nodes. By applying voltages to the boundary nodes, we obtain a unique harmonic extension on all vertices

voltages, which might be found out by the Ohm's and Kirchhoff's laws. Investigating various properties of these harmonic extensions has given rise to many combinatorial objects, such as electrical response matrices, effective resistances, spanning trees, and groves. These objects have appeared in various mathematical theories, from Potts models (see zero Potts models [6]) to Abelian sandpile models [4].

The focus of my talk will be on the theory of planar circular electrical networks, which is closely related to the geometry of Lagrangian and Isotropic Grassmannian [2], [3], [7]. We will present two explicit constructions [2], [3] for the embedding of electrical networks to the non-negative part of the Isotropic Grassmannian using their response matrices and effective resistance matrices, respectively. Using the first construction, we will demonstrate a sketch of a new cluster solution to the discrete version of the Calderón problem, which is also known as the inverse problem of electrical impedance tomography [1]. Using the second one, we will provide a characterization of the resistance distance, which is widely used in chemistry and phylogenetic network theory [5].

The author was supported by the Russian Science Foundation grant 20-71-10110 (P).

[1] Borcea, L., Druskin, V., Vasquez, F. G.,``Electrical impedance tomography with resistor networks. Inverse Problems'', Vol.24, No.3, (2008).

[2] Bychkov, B., Gorbounov, V., Guterman, L., Kazakov, A.,``Symplectic geometry of electrical networks'', Journal of Geometry and Physics, Vol.207, (2025).

[3] Bychkov B.,  Gorbounov V.,  Kazakov A.,  Talalaev D., ``Electrical Networks, Lagrangian Grassmannians, and Symplectic Groups,'' Moscow Mathematical Journal, Vol.23, No.2, (2023).

[4] Dhar, D., ``The abelian sandpile and related models'', Physica A: Statistical Mechanics and its applications, Vol.263, No. 1-4., (1999).

[5] Forcey, S., ``Circular planar electrical networks, split systems, and phylogenetic networks'', SIAM Journal on Applied Algebra and Geometry, Vol.7, No. 1, (2023).

[6] Fortuin, C. M., Kasteleyn, P. W.,``On the random-cluster model: I. Introduction and relation to other models'', Physica, Vol. 57, No. 4., (1979).

[7] Lam T., ``Totally nonnegative Grassmannian and Grassmann polytopes,'' arXiv preprint arXiv:1506.00603, (2015).

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

Speaker: D. Talalaev (MSU,YarSU)

Title:  RE-algebras, Gelfand-Retakh quasi-determinants, and quantization of the complete Toda system

Date and time:  30.10.2024, 17:00 (GMT +03:00)

Abstract: In 1991, Gelfand and Retakh proposed the technique of quasi-determinants, generalizing the long-known concept of Schur complement, at the same time they described the commutative subalgebra in RTT algebra using some quasi-determinants. In my report, I will talk about a similar construction in the case of the reflection equation algebras (RE-algebras), using an alternative family of quasi-determinants, which is apparently indicates the alternative structure of the RE-algebra as a quantum homogeneous space over the quantum group. The resulting family is, among other things, a quantization of the complete Toda system for the case of a RE-algebra.

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

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