Speaker: D. Talalaev (MSU,YarSU)

Title: On a family of Poisson brackets on gl(n) compatible with the Sklyanin bracket

Date and time:  23.04.2025, 17:00 (GMT +03:00)

Abstract:  The talk is focused on the family of compatible quadratic Poisson brackets on gl(n), generalizing the Sklyanin one. For any of the brackets in the family, the argument shift determines the compatible linear bracket. I will describe the application of the bi-Hamiltonian formalism for some pencils from this family, namely a method for constructing involutive subalgebras for a linear bracket starting by the center of the quadratic bracket. I will provide some interesting examples of families of this type.

An important ingredient of the construction is the family of antidiagonal principal minors of the Lax matrix. A crucial but quite unbiguous condition of the log-canonicity of  brackets of these minors with all the generators of the Poisson algebra establishes a relation of our families with cluster algebras, a similar property arises in the context of Poisson structures consistent with mutations.

The talk is based on the recent joint paper with V.V. Sokolov https://arxiv.org/abs/2502.16925

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

Speaker: Pavlos Kassotakis (University of Patras, Greece)

Title: On  pentagon and entwining tetrahedron maps

Date and time:  9.04.2025, 17:00 (GMT +03:00)

Abstract:  In this talk, we present equivalence classes of rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, up to  Möbius transformations, we find quadrirational one-component maps of rational functions in two arguments that serve as solutions of the pentagon equation. Also, provided that a pentagon map admits at least one partial inverse, we obtain  entwining pentagon set theoretical solutions.  Furthermore, we show how to obtain Yang–Baxter and entwining tetrahedron maps from pentagon maps.

Video of the talk
https://rutube.ru/video/9a48e95f99c8b4f2c004d3d7f129be74/

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

Speaker: Vsevolod Gubarev (Sobolev Institute of Mathematics; Novosibirsk State University)

Title: Rota-Baxter operators and Yang-Baxter equation

Date and time:  26.03.2025, 17:00 (GMT +03:00)

Abstract:  In 1960, G. Baxter introduced the notion of Rota-Baxter operator. In 1960-70s, such operators were actively studied on Banach algebras. In 1980s, A.A. Belavin, V.G. Drinfeld and M.A. Semenov-Tian-Shansky rediscovered Rota-Baxter operators while studying classical and modified Yang-Baxter equations. In 2000, M. Aguiar found even deeper connection between Rota-Baxter operators defined on associative algebras and associatie Yang-Baxter equation. In 2020, Rota-Baxter operators on a group were defined by L. Guo, H. Lang, and Y. Sheng, such operators are connected to quantum Yang-Baxter equation.

Video of the talk
https://rutube.ru/video/d0e4f0038b05ce289f5543a976cff2db/

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

Speaker: Boris Bychkov (UHaifa, HSE)

Title: KP integrability in topological recursion (continuation)

Date and time:  12.03.2025, 17:00 (GMT +03:00)

Abstract:  Topological recursion of Chekhov-Eynard-Orantin is a remarkable universal recursive procedure that has been found in many enumerative geometry problems, from combinatorics of maps, to random matrices, Gromov-Witten invariants, Hurwitz numbers, Mirzakhani's hyperbolic volumes of moduli spaces, knot polynomials. A recursion needs an initial data: a spectral curve, and the recursion defines the sequence of invariants of that spectral curve. 

Kadomtsev-Petviashvili hierarchy is an integrable hierarchy of nonlinear PDEs. Except for many important properties, it quite often appears in the applications: a lot of functions from combinatorics, mathematical physics, theory of moduli spaces and Gromov-Witten theory are solutions to the KP hierarchy.

In the talk I will define the KP integrability property for the topological recursion invariants and show that TR invariants are KP integrable if and only if the corresponding spectral curve is rational. If time permits I will discuss the construction of the KP tau function on the TR spectral curve of any genus which can be seen as a non-perturbative generalization of the Krichever's construction of the KP tau function on any elliptic curve.

The talk is based on the series of joint works with A. Alexandrov, P. Dunin-Barkowski, M. Kazarian and S. Shadrin ( https://arxiv.org/abs/2309.12176,  https://arxiv.org/abs/2406.07391,  https://arxiv.org/abs/2412.18592).

Video of the talk
https://rutube.ru/video/d4e0a34115b7cafd749b0baee99840c0/

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

Speaker: Boris Bychkov (UHaifa, HSE)

Title: KP integrability in topological recursion

Date and time:  26.02.2025, 17:00 (GMT +03:00)

Abstract:  Topological recursion of Chekhov-Eynard-Orantin is a remarkable universal recursive procedure that has been found in many enumerative geometry problems, from combinatorics of maps, to random matrices, Gromov-Witten invariants, Hurwitz numbers, Mirzakhani's hyperbolic volumes of moduli spaces, knot polynomials. A recursion needs an initial data: a spectral curve, and the recursion defines the sequence of invariants of that spectral curve. 

Kadomtsev--Petviashvili hierarchy is an integrable hierarchy of nonlinear PDEs. Except for many important properties, it quite often appears in the applications: a lot of functions from combinatorics, mathematical physics, theory of moduli spaces and Gromov-Witten theory are solutions to the KP hierarchy.

In the talk I will define the KP integrability property for the topological recursion invariants and show that TR invariants are KP integrable if and only if the corresponding spectral curve is rational. If time permits I will discuss the construction of the KP tau function on the TR spectral curve of any genus which can be seen as a non-perturbative generalization of the Krichever's construction of the KP tau function on any elliptic curve.

The talk is based on the series of joint works with A. Alexandrov, P. Dunin-Barkowski, M. Kazarian and S. Shadrin ( https://arxiv.org/abs/2309.12176,  https://arxiv.org/abs/2406.07391,  https://arxiv.org/abs/2412.18592).

Video of the talk
https://rutube.ru/video/1e8efa4c69b46e4c98c44e4cac03e6f4/

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

Speaker: V.V. Sokolov (MIPT, Moscow)

Title: Integrable vector evolution equations

Date and time:  15.01.2025, 17:00 (GMT +03:00)

Abstract: We consider evolution equations of the form $U_t=\sum_{i=1}^n a_i U_i$, where $U(x,t)$ is a vector of arbitrary size, $U_i=\frac{\partial^i U}{\partial x ^ i}$ and the coefficients $a_i$ are functions of scalar products $(U_i, U_j)$. The symmetry approach to classification of scalar evolution equations is generalised to the case of such vector isotropic evolution equations.  A further generalisation to  the anisotropic case is proposed. Some examples and classification results are presented.

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

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