Speaker: Boris Bychkov (University of Haifa)

Date and time:  1.03.2023, 17:00 (GMT +03:00)

Title: Topological recursion for generalized double Hurwitz numbers

Abstract: Topological recursion 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. 

In the talk I will define the topological recursion, spectral curves and their invariants; I will describe our results on explicit closed algebraic formulas for generating functions of generalized double Hurwitz numbers, and how this allows to prove topological recursion for a wide class of problems.

If time permits I'll talk about the implications for the so-called ELSV-type formulas (relating Hurwitz-type numbers to intersection numbers on the moduli spaces of algebraic curves).

The talk is based on the series of joint works with P. Dunin-Barkowski, M. Kazarian and S. Shadrin.

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

Speaker: Alexey Basalaev (HSE, Skoltech)

Date and time:  15.02.2023, 17:00 (GMT +03:00)

Title: Integrable systems associated to infinite families of Dubrovin-Frobenius manifolds

Abstract: It was an idea of B.Dubrovin to study integrable systems using the theory of Dubrovin-Frobenius manifold. Currently there are two constructions allowing one to associate an integrable system to a Dubrovin-Frobenius manifold - DZ and DR hierarchies. Both constructions are rather complicated and not easy to compute. With both constructions one gets Gelfand-Dikij hierarchy starting from a type A Dubrovin-Frobenius manifold. In this talk we will present another construction that is not as general as DR and DZ. constructions but really easy to compute. We will show that our construction produces KP hierarchy from type A Dubrovin-Frobenius manifolds and discuss the cases of type D Dubrovin-Frobenius manifolds and open extensions coming from the open Gromov-Witten theories.

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

Speaker: Folkert Müller-Hoissen (Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany)

Date and time: 18.01.2023, 17:00 Moscow time (GMT +03:00)
Title: Higher Bruhat and Tamari orders, simplex and polygon equations

Abstract: We first present an introduction to Bruhat and higher Bruhat (partial) orders, highlighting the very simple underlying ideas. Higher Bruhat orders have been introduced by Manin and Schechtman in 1986, who also revealed them as the crucial structure behind the hierarchy of simplex equations, of which the famous Yang-Baxter equation and the tetrahedron (Zamolodchikov) equation are first members.
Via a certain decomposition of the higher Bruhat orders we arrive at "higher Tamari orders", as defined in a joint work with Aristophanes Dimakis (Tamari Memorial Festschrift, Progress in Mathematics, vol. 299, 2012, pp. 391-423), where this structure arose from an exploration of a class of soliton solutions of the Kadomtsev-Petviashvili (KP) equation, which form rooted binary trees at fixed time. We had conjectured that these higher Tamari orders are equivalent to what was known as higher Stasheff-Tamari orders (defined in terms of triangulations of cyclic polytopes), and this has recently been proved by Nicholas Williams (arXiv:2012.10371).
Following my work with Dimakis, SIGMA 11 (2015) 042, we explain how in the same way as the higher Bruhat orders determine the hierarchy of simplex equations, the higher Tamari orders determine a hierarchy of "polygon equations", of which the famous pentagon equation is a member.  

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

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