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

Speaker:  Vladimir Dragović (UT Dallas)

Date and time:  14.07.2021, 17:00 (GMT +03:00)

Title: Integrable billiards, extremal polynomials, and combinatorics

Abstract:  A comprehensive study of periodic trajectories of the billiards within ellipsoids in the d-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of d intervals on the real line. Classification of periodic trajectories is based on a new combinatorial object: billiard partitions.

The case study of trajectories of small periods T, d ≤ T ≤ 2d is given. In particular, it is shown that all d-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates d + 1-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for d = 3.

The talk is based on the following papers:

V. Dragović, M. Radnović, Periodic ellipsoidal billiard trajectories and extremal polynomials, Communications. Mathematical Physics, 2019, Vol. 372, p. 183-211.

G. Andrews, V. Dragović, M. Radnović, Combinatorics of the periodic billiards within quadrics, arXiv: 1908.01026, The Ramanujan Journal, DOI: 10.1007/s11139-020-00346-y.

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

Speaker:  Adam Doliwa (University of Warmia and Mazury, Poland)

Date and time:  30.06.2021, 17:00 (GMT +03:00)

Title: Non-commutative birational maps satisfying Zamolodchikov's tetrahedron equation from projective geometry over division rings II

Abstract:  The report will be a continuation of the talk on June 16, which already introduced the general concepts of compatibility conditions in higher dimensions and the interpretation of the corresponding maps in terms of Desargues configurations   https://cis.uniyar.ac.ru/node/369

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

Speaker:  Adam Doliwa (University of Warmia and Mazury, Poland)

Date and time:  16.06.2021, 17:00 (GMT +03:00)

Title: Non-commutative birational maps satisfying Zamolodchikov's tetrahedron equation from projective geometry over division rings

Abstract:  The notion of multidimensional consistency is an important ingredient of the contemporary theory of integrable systems. In my talk I will focus on geometric origin of the multidimensional consistency of Hirota's discrete KP equation. Because the relevant geometric theorem is valid in projective geometries over division rings, we are led to non-commutative version of the equation, which is due to Nimmo. I will show how four-dimensional consistency of the discrete KP system gives the corresponding solution to Zamolodchikov's tetrahedron equation (generalization of the Yang-Baxter equation to more dimensions). In particular, different algebraic descriptions of the same geometric theorem lead to different (but of course equivalent) solutions of the equation. Finally, I will discuss how natural ultra-locality condition imposed on the solution gives Weyl commutation relations. The talk is based on joint works with Sergey Sergeev and Rinat Kashaev. 

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