Speaker: Dmitry Talalaev (MSU, YarSU, ITEP)

Date and time:  14.12.2022, 17:00 (GMT +03:00)

Title: The full Toda system, QR decomposition and geometry of the flag varieties

Abstract: The full Toda system is a generalization of an open Toda chain, which is one of the archetypal examples of integrable systems. The open Toda chain illustrates the connection of the theory of integrable systems with the theory of Lie algebras and Lie groups, is a representative of the Adler-Kostant-Symes scheme for constructing and solving such systems. Until recently, only some of the results from this list were known for the full Toda system. I will talk about the construction, the commutative family, quantization and solution of the full Toda system by the QR decomposition method, as well as about the application of this system to the geometry of flag vaireties. The material of my talk is based on several joint works with A. Sorin, Yu. Chernyakov and G. Sharygin.

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

Speaker: Valeriy Bardakov (Sobolev Institute of Mathematics, Novosibirsk)

Date and time: 30.11.2022, 17:00 (GMT +03:00)

Title:  REPRESENTATIONS OF THE VIRTUAL BRAID GROUP AND THE YANG-BAXTER EQUATION

Abstract: It is well known that a solution for the Yang–Baxter equation (YBE) or that is equivalent for the braid equation (BE) gives a representation of the braid group Bn. In this talk I explain a connection between YBE and representations of the virtual braid group VBn. In particular, I show that any solution (X, R) for the Yang–Baxter equation with invertible R defines a representation of the virtual pure braid group VPn, for any n ≥ 2, into Aut(X⊗n) for linear solution and into Sym(Xn) for set-theoretic solution. Any solution of the BE with invertible R gives a representation of a normal subgroup Hn of VBn. As a consequence of two these results we get that any invertible solution for the BE or YBE gives a representation of VBn.

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

Speaker: Dmitry Talalaev (MSU, YarSU, ITEP)

Date and time:  2.11.2022, 17:00 (GMT +03:00)

Title:  Extension of braided sets and higher generalizations

Abstract: A braided set is a synonym for a solution of the set-theoretic Yang-Baxter equation. It is important to rephrase this in a categorical language from the point of view of natural questions of morphisms, extensions and simple objects in this family. I will tell about several results in the problem of constructing extensions of braided sets and how this problem can be generalized to 2-braided categories, how to build extensions of sets with solutions of the Zamolodchikov tetrahedron equation.

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

Speaker:  Alexandr Orlov (Institute of Oceanology, RAS)

Date and time: 19.10.2022, 17:00 (GMT +03:00)

Title: Graphs, Hamiltonian systems and Hurwitz numbers

Abstract: The quantum Calogero equation at the point of free fermions can be considered as an equation written in terms of the eigenvalues of some matrix X, and the Hamiltonian can be written as the Casimir operator, an element of the center of the universal enveloping algebra. This equation also makes sense of the generalized Mironov-Morozov-Natanzon cut-an-join equation (it generalizes the Guldon-Jackson cut-and-join equation, which describes the composition of a transposition with an element of a symmetric group in terms of the action of a differential operator on a polynomial of many variables). I will consider generalizations of this equation obtained for a set of matrices $X_1,\dots,X_n$, and classes of their solutions constructed using bipartite graphs, and also tell you about the connection of these problems with the enumeration of coverings of the Riemann surface with an embedded graph (Hurwitz numbers).

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

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

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

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

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

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