Speaker: Leonid Chekhov (Steklov Mathematical Institute and Michigan State University)

Title: Symplectic groupoid: geometry, networks, and moduli spaces of closed Riemann surfaces

Date and time:  25.06.2025, 17:00 (GMT +03:00)

Abstract: I will describe the Bondal's symplectic groupoid: a set of pairs (B,A) with A unipotent upper-triangular matrices and B an element of GL(n) being such that the matrix B A B^T is itself unipotent upper triangular. Since works of J.Nelson, T.Regge, B.Dubrovin,  and M.Ugaglia it was known that entries of A can be identified with geodesic functions on a Riemann surface with holes; these entries then enjoy a closed Poisson algebra (reflection equation) expressible in the r-matrix form. In our recent work with M.Shapiro, we solved the symplectic groupoid in terms of planar networks; we used this solution to construct a complete set of geodesic functions for a closed Riemann surface of genus 2; all geodesic functions are elements of the upper cluster algebra whereas Dehn twists are described by cluster mutations. This is a joint work with M.Shapiro.

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

Speaker: G.I. Sharygin (MSU, MIPT)

Title: Symmetries of the full symmetric Toda system and Li-Bianchi integrability

Date and time:  11.06.2025, 17:00 (GMT +03:00)

Abstract: The full symmetric Toda system is an integrable Hamiltonian system on the space of symmetric real matrices, similar to the open Toda chain. In this report, I will talk about how to build vector fields that preserve this system. In particular, it follows that this system is integrable in the sense of the Li-Bianchi theorem (that is, it has a solvable algebra of symmetries of maximum dimension).

Video of the talk
https://rutube.ru/video/private/7a03bddb8b8e25ca6c2bbb261532c550/?p=UtuEJV79owNeV2L6a2uh5w

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

Speaker: Pavlos Xenitidis (Liverpool Hope University, UK)

Title: Noncommutative discrete KdV equations, their symmetries and reductions

Date and time:  21.05.2025, 17:00 (GMT +03:00)

Abstract: Employing the Lax pairs of the discrete noncommutative Hirota's Korteweg-de Vries (KdV) and the potential KdV equations, we derive differential-difference equations consistent with these equations which play the role of generalised symmetries of the latter. Miura transformations map them to a noncommutative modified Volterra equation and its master symmetry are given. The use of the symmetries for the reduction of the potential KdV equation is demonstrated and the reductions to a noncommutative discrete Painleve equation and a system of partial differential equations generalising the Ernst equation and the Neugebauer-Kramer involution are presented. A Darboux and an auto-Backlund transformation for the Hirota KdV are presented and their relation to the noncommutative Yang-Baxter map is given.

Video of the talk:
https://rutube.ru/video/private/ed06ef18476635b102f18858696669de/?p=mAE0iTB1Ur5bM05L5hRtpQ

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

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

Video of the talk
https://rutube.ru/video/01eb3527f465571105f0ed1d5a93b716/

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