Speaker: Pavlos Kassotakis (University of Kent, UK; Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, Russia)

Date and time: 27.04.2022, 17:00 (GMT +03:00)

Title: Hierarchies of compatible maps and integrable difference systems

Abstract: We introduce families of non-Abelian compatible maps associated with Nth order discrete spectral problems. In that respect we have hierarchies of families of compatible maps that in turn are associated with hierarchies of set-theoretical solutions of the 2-simplex equation a.k.a Yang-Baxter maps. These hierarchies are naturally associated with integrable difference systems with variables defined on edges of an elementary cell of the $\mathbb{Z}^2$ graph, that in turn lead to hierarchies of difference systems with variables defined on vertices of the same cell. Furthermore, these hierarchies with vertex variables are point equivalent with the explicit form of what will be called non-Abelian lattice-NQC(Nijhoff-Quispel-Capel) Gel'fand-Dikii hierarchy.

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

Speaker: P. Grinevich (Moscow State University, MIAN)

Date and time: 13.04.2022, 17:00 (GMT +03:00)

Title: Signatures on plabic graphs and completely non-negative Grassmannians

Abstract: As shown by A. Postnikov the cells of completely non-negative Grassmannians can be rationally parametrized by graphs embedded in a disk with positive weights on the edges. In this case the matrix elements representing the Grassmannian points are given as sums along all possible paths from the boundary sources to the boundary sinks. An alternative approach is to define the Grassmannian points by solving a system of linear equations corresponding to the vertices of the graph. In this case positivity is achieved only with the correct choice of signs on the edges called a signature. T. Lam proved the existence of a signature consistent with the property of complete positivity without presenting it explicitly. We give an explicit construction and prove the uniqueness of such a signature up to the action of the natural gauge group.

The report is based on joint work with S. Abenda.

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

Speaker: Andrea Caranti (University of Trento, Italy)

Date and time: 30.03.2022, 17:00 (GMT +03:00)

Title: Regular subgroups, skew braces, gamma functions and Rota–Baxter operators

Abstract: Skew braces, a novel  algebraic structure introduced only in 2015, have already spawned a sizable literature. The skew  braces with a  given additive group structure  correspond to the  regular  subgroups  of  the permutational  holomorph  of  such  a group. These regular subgroups can in  turn be described in  terms of certain so-called gamma  functions from the group  to its automorphism group, which are characterised by a functional equation. We will show how gamma functions  can be used in studying skew braces, underlining in particular their relationship to Rota-Baxter operators.

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

Speaker: Sofya Mukhamedjanova (Kazan Federal University)

Date and time: 16.03.2022, 17:00 (GMT +03:00)

Title: Explicit formulas for chromatic polynomials of some series-parallel graphs 

Abstract:  The main goal of my report is to present explicit formulas for chromatic polynomials of some planar series-parallel graphs (sp-graphs). The necklace-graph considered in this report is the simplest non-trivial sp-graph. We have provided the explicit formula for calculating the chromatic polynomial of common sp-graphs. In addition, we have presented the explicit formulas for calculating chromatic polynomials of the ring of the necklace graph and the necklace of the necklace graph. Chromatic polynomials of the necklace graph and the ring of the necklace graph have been initially obtained by transition to the dual graph and the subsequent using of the flow polynomial. The use of the partition function of the Potts model is a more general way to evaluate chromatic polynomials. In this method, we have used the parallel- and series-reduction identities that were introduced by A. Sokal. We have developed this idea and introduced the transformation of the necklace-graph reduction. Using this transformation makes it easier to calculate chromatic polynomials for the necklace-graph, the ring of the necklace graph, as well as allows to calculate the chromatic polynomial of the necklace of the necklace graph. This report is based on our joint work E. Yu. Lerner, S. A. Mukhamedjanova, “Explicit formulas for chromatic polynomials of some series-parallel graphs” http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=uzku&paperid=1459&option_lang=rus

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

Speaker: E.V. Ferapontov (Loughborough University)

Date and time: 02.02.2022, 17:00 (GMT +03:00)

Title: Second-order integrable Lagrangians and WDVV equations

Abstract: I will discuss integrability of Euler-Lagrange equations associated with 2D and 3D second-order Lagrangians. By deriving integrability conditions for the Lagrangian density, examples of integrable Lagrangians expressible via elementary functions, elliptic functions and modular forms are constructed.  Explicit link of second-order integrable Lagrangians to the WDVV equations is also established. 

Based on joint work with Maxim Pavlov and Lingling Xue: Second-order integrable Lagrangians and WDVV equations, Lett. Math. Phys. (2021) 111:58; arXiv:2007.03768.

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

Speaker: Pavlos Xenitidis (Liverpool Hope University)

Date and time: 22.12.2021, 17:00 (GMT +03:00)

Title: Darboux and Bäcklund transformations for integrable difference equations

Abstract: Motivated by known results on integrable differential equations, I will discuss Darboux and Bäcklund transformations for integrable difference equations. More precisely I will present a method for  the construction of these transformations, derive their superposition principle, explain the relation of the latter to Yang-Baxter maps, and demonstrate their implementation in the construction of solutions. In this talk I will use two  illustrative examples, namely the Hirota KdV equation and an integrable discretisation of the NLS equation (aka Adler-Yamilov system), and I will discuss the extension of these ideas to noncommutative systems.

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

Speaker:  Nikolay Erochovets (Moscow State University)

Date and time:  8.12.2021, 17:00 (GMT +03:00)

Title: Canonical geometrization of orientable 3-manifolds defined by vector-colourings of 3-polytopes

Abstract:  In short geometrization conjecture of W.Thurston (finally proved by G.Perelman) says that any oriented 3-manifold can be canonically partitioned into pieces, which have a geometric structure of one of the eight types.

In the seminal paper (1991) M.W.Davis and T.Januszkiewicz introduced a wide class of n-dimensional manifolds -- small covers over simple n-polytopes.

We give a complete answer to the following problem: to build an explicit canonical decomposition for any orientable 3-manifold defined by a vector-colouring of a simple 3-polytope, in particular for a small cover.

The proof is based on analysis of results in this direction obtained before  by different authors.

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

Speaker:  Boris Bychkov (Higher School of Economics, Yaroslavl State University)

Date and time:  10.11.2021, 17:00 (GMT +03:00)

Title: Electrical networks and Lagrangian Grassmannians

Abstract:  Electircal network is a graph in a disk with positive weights on edges. The (compactified) space $E_n$ of electrical networks embeds into the totally nonnegative Grassmannian $\mathrm{Gr}{\geq 0}(n-1,2n)$. I will talk about the new parametrisation of $E_n$ which defines an embedding into the Grassmannian $\mathrm{Gr}(n-1,V)$, where $V$ is a certain subspace of dimension $2n-2$ and moreover into the nonnegative Lagrangian Grassmannian $\mathrm{LG}{\geq 0}(n-1)\subset\mathrm{Gr}(n-1,V).$

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

Speaker:  Tassos Bountis (P.G. Demidov Yaroslavl State University, Russia)

Date and time:  27.10.2021, 17:00 (GMT +03:00)

Title: Integrable and nonintegrable Lotka-Volterra systems

Abstract:  In recent years, there has been renewed interest in the study of anti-symmetric Lotka-Volterra Hamiltonian (LVH) systems of competing species. In particular, it is interesting to add linear (or nonlinear) terms to these systems, and either seek to preserve integrability, or investigate the dynamics of ''nearby'' nonintegrable systems in the n-dimensional phase space.

In this talk, I will first show how new integrable classes of LVH systems were discovered applying the Painlevé property, and then demonstrate that ''nearby'' non-integrable systems typically continue to possess very simple dynamics. Finally, I will discuss some very recent findings revealing possible connections between the Painlevé property and Brenig's method of integrating polynomial systems of ODEs by reduction to canonical form.

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

Speaker:  Alexander Shapiro (University of Edinburgh, Great Britain)

Date and time:  29.09.2021, 17:00 (GMT +03:00)

Title: Cluster structure on quantum Toda chain

Abstract:  Double Bruhat cells serve as phase spaces for a family of (degenerately) integrable systems, the generalized open Toda lattices. From cluster algebraic perspective double Bruhat cells were studied by Berenstein–Fomin–Zelevinsky and Fock–Goncharov, while the Toda lattice was investigated by Gekhtman–Shapiro–Vainshtein. In this talk I will describe cluster Poisson coordinates on double Bruhat cells, and use them to recover the quantum Toda lattice. If time permits, I will also briefly discuss our joint work with Gus Schrader on the eigenfunctions of the quantum Toda Hamiltonians.

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