Интегрируемые системы

Seminar (online): M. Pavlov "Associativity Equations. Frobenius Manifolds. B.A. Dubrovin's construction"

Submitted by A.Tolbey on Sat, 10/01/2022 - 19:17

Speaker: Maxim Pavlov (LPI RAS)

Date and time: 5.10.2022, 17:00 (GMT +03:00)

Title:  Associativity Equations. Frobenius Manifolds. B.A. Dubrovin's construction

Abstract: The system of associativity equations arises in Topological Field Theory. B.A. Dubrovin drew attention to it in a series of works 1992-1994. It describes orthogonal curved Egorov coordinate systems related to the Egorov semi-Hamiltonian systems of hydrodynamic type integrable by the generalized Tsarev method. Even more relevant are hydrodynamic-type systems having a local Hamiltonian structure associated with the Dubrovin-Novikov Poisson bracket. The presence of two such matched Poisson brackets generates a Frobenius manifold.

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

Event date
Wed, 10/05/2022 - 17:00

Seminar (online): A. Buryak "Moduli spaces of curves and classification of integrable PDE"

Submitted by A.Tolbey on Tue, 09/13/2022 - 21:55

Speaker: Alexandr Buryak (HSE)

Date and time: 21.09.2022, 17:00 (GMT +03:00)

Title: Moduli spaces of curves and classification of integrable PDE

Abstract: We consider integrable evolutionary PDEs with one spatial variable, by integrability we understand the presence of an infinite family of infinitesimal symmetries. I will tell about some classification problems of such PDEs which are purely algebraic and even elementary in their formulation. Surprisingly, the hypothetical solution is absolutely not elementary and is related with the geometry of the moduli spaces of algebraic curves.

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

Event date
Wed, 09/21/2022 - 17:00

Seminar (online): G.I. Sharygin "Noncommutative Painleve 4 equation"

Submitted by A.Tolbey on Wed, 06/15/2022 - 08:16

Speaker: G. Sharygin

Date and time: 22.06.2022, 17:00 (GMT +03:00)

Title: Noncommutative Painleve 4 equation

Abstract: Let $R$ be a unital division ring, equipped with a differentiation $D$; if $t\in R$ verifies the equation $Dt=1$, we can regard $D$ as the operator $\frac{d}{dt}$. We develop a theory of Painleve 4 equation in $R$ with $t$ playing the role of time variable. It turns out that the classical results like B\"acklund transformations, Hamiltonian and symmetric forms of this equation, Hankel determinant formulas for the solutions etc. have noncommutative analogs in this case; in particular, one has to use Gelfand and Retakh theory of quasi-determinants instead of the usual Hankel determinants and deal with different types of Painleve 4 equation, depending on the formulas we choose. The talk is based on a joint work with I. Bobrova, V. Retakh and V. Rubtsov.

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

Event date
Wed, 06/22/2022 - 17:00

Seminar (online): G. Koshevoy "On Manin-Schechtman orders related to directed graphs"

Submitted by A.Tolbey on Wed, 06/01/2022 - 14:22

Speaker:  Gleb Koshevoy (CIEM RAS, Moscow)

Date and time: 8.06.2022, 17:00 (GMT +03:00)

Title: On Manin-Schechtman orders related to directed graphs

Abstract: Studying higher simplex equations (Zamolodchikov equations), in 1989 Manin and Schechtman introduced the notion of a higher Bruhat order on the $d$-element subsets of a set $[n]=\{1,2,\ldots,n\}$. Among other results in this field, they proved that the set of such orders for $n,d$ fixed, endowed with natural local transformations, constitutes a poset with one minimal and one maximal elements.

I will report on a wider model, involving the so-called convex order on certain path systems in an acyclic directed graph.

We introduce local transformations, or flips, on such orders and prove that the resulting structure gives a poset with one minimal and one maximal elements as well, and get relations between convex orders in neighboring dimensions, yielding a generalization of the above-mentioned classical results. This is  a joint work with V. Danilov and A. Karzanov.

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

Event date
Wed, 06/08/2022 - 17:00

Seminar (online): P. Kassotakis "Hierarchies of compatible maps and integrable difference systems"

Submitted by A.Tolbey on Fri, 04/22/2022 - 21:31

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

Event date
Wed, 04/27/2022 - 17:00

Seminar (online): P. Grinevich "Signatures on plabic graphs and completely non-negative Grassmannians"

Submitted by A.Tolbey on Wed, 04/06/2022 - 19:50

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

Event date
Wed, 04/13/2022 - 17:00

Seminar (online): A. Caranti "Regular subgroups, skew braces, gamma functions and Rota–Baxter operators"

Submitted by A.Tolbey on Wed, 03/23/2022 - 20:49

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

Event date
Wed, 03/30/2022 - 17:00

Seminar (online): S. Mukhamedjanova "Explicit formulas for chromatic polynomials of some series-parallel graphs"

Submitted by A.Tolbey on Thu, 03/10/2022 - 10:22

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

Event date
Wed, 03/16/2022 - 17:00

Seminar (online): E.V. Ferapontov "Second-order integrable Lagrangians and WDVV equations"

Submitted by A.Tolbey on Sun, 01/30/2022 - 17:02

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

Event date
Wed, 02/02/2022 - 17:00

Seminar (online): P. Xenitidis "Darboux and Bäcklund transformations for integrable difference equations"

Submitted by A.Tolbey on Wed, 12/15/2021 - 22:21

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

Event date
Wed, 12/22/2021 - 17:00