Speaker: Gurevich Dimitry (IITP, Moscow)
Date and time:  24.05.2023, 17:00 (GMT +03:00)
Title:  Reflection Equation Algebra and related combinatorics

Abstract: Reflection Equation Algebras constitute a subclass of the so-called  Quantum Matrix algebras. Each of the REA is associated with a quantum $R$-matrix. In a sense the REA corresponding to quantum R-matrix of Hecke type can be considered as q-counterparts of the commutative algebra Sym(gl(N)) or the enveloping algebras U(gl(N)) depending on its realization. On any such RE algebra there exist analogs of some symmetric polynomials, namely  the power sums and the Schur functions.

In my talk I plan to exhibit q-versions  of the Capelli formula, the Frobenius formula, related to  this combinatorics. Also I plan to introduce analogs of the Casimir operators and  perform  their spectral analysis.

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

Speaker: Evgeny Smirnov (HSE, IUM)

Date and time:  10.05.2023, 17:00 (GMT +03:00)

Title: Polytopes and K-theory of toric and flag varieties

Abstract: In 1992 Askold Khovanskii and Alexander Pukhlikov proposed a description of the cohomology ring for a smooth toric variety as the quotient of the ring of differential operators with constant coefficients modulo the annihilator of the volume polynomial for the moment polytope of this variety. Later Kiumars Kaveh observed that the cohomology ring of a full flag variety can be obtained by applying the same construction to Gelfand-Zetlin polytope.

I will speak about our work with Leonid Monin generalizing these results for the case of K-theory. Namely, we describe algebras with a Gorenstein duality pairing as quotients of the ring generated by shift operators. Then we apply this construction to describe the Grothendieck ring of a smooth toric variety; for this we consider shift operators modulo the annihilator of the Ehrhart polynomial of the moment polytope (this substitutes the volume polynomial). Finally, this construction can be generalized to the case of full flag varieties of type A. This description allows us to make computations in the Grothendieck ring of a full flag variety by intersecting faces of Gelfand-Zetlin polytopes; this generalizes our result with Valentina Kiritchenko and Vladlen Timorin.

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

Speaker: Maxim Pavlov, Lebedev Physical Institute, Moscow, Russia
Date and time:  26.04.2023, 17:00 Moscow time (17:00 Helsinki time, 16:00 Berlin time and 15:00 London time). 
Title: Extended KP hierarchy and its two-dimensional reductions
Abstract: Conventionally, the KP hierarchy can be divided into several parts:

1. purely differential part (obtained from the compatibility conditions of differential operators);
2. pseudo-differential part (obtained from the compatibility conditions of pseudo-differential operators);
3. semi-discrete part (obtained from the compatibility conditions of differential and difference operators);
4. purely discrete part (obtained from the compatibility conditions of difference operators).

The restriction of these operators to the stationary case leads to two-dimensional reductions.
Thus, three-dimensional equations will be constructed from the KP hierarchy, as well as their two-dimensional reductions.

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

Speaker: Georgy Sharygin (MSU, ITEP, MIPT)

Date and time:  12.04.2023, 17:00 (GMT +03:00)

Title: Quasi-derivations of Ugl_n and the quantum argument shift method

Abstract: In my talk I will tell, how one can use the "quasi-derivations" to partially transfer the "argument shift method" (a method used to obtain commutative subalgebras in Poisson algebras) to the universal enveloping algebra Ugl_n. "Quasi-derivations" of Ugl_n, is a set of linear operators on Ugl_n, constructed earlier by Gurevich, Pyatov and Saponov with the help of reflection equation algebra; they also have many other definitions. I will speak about the properties of these operators and then show that the iterations of these operators on the generators of the center of Ugl_n commute with each other.

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

Speaker: Jarmo Hietarinta (Department of Physics and Astronomy, University of Turku, Finland)

Date and time:  29.03.2023, 17:00 (GMT +03:00)

Title: Evolution equations, lattice equations and the three soliton condition

Abstract: This is an overview talk. We will discuss the connections between evolution equations having soliton solutions and lattice

equations. In particular we look at the property of integrability, and it seems that the existence of a "three-soliton solution" is an

important and common underlying property. As a concrete example we take the Korteweg-de Vries equation, continuous and discrete.

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

Speaker: Sotiris Konstantinou-Rizos (P.G. Demidov Yaroslavl State University, Russia)
Date and time: 15.03.2023, 17:00 (GMT +03:00)

Title: Correspondences and N-simplex maps on groups and rings

The Yang--Baxter equation and Zamolodchikov's tetrahedron equation are two of the most fundamental equations of Mathematical Physics, and they are members of the general family of n-simplex equations for n=2 and n=3, respectively. The study of n-simplex maps, namely set-theoretical solutions to the n-simplex equation, was formally initiated by Drinfeld for n = 2.

In this talk, I will present new methods for constructing new solutions to 3- and 4-simplex equations. I will demonstrate the role of correspondences for deriving new n-simplex maps which do not belong to any of the known classification lists. Next, I will present some new tetrahedron maps on groups and rings. Moreover, I will show a method for constructing nontrivial 4-simplex extensions of tetrahedron maps. Finally, I will present some new n-simplex maps.

This talk is mainly based on the following papers:

[1] S. Konstantinou-Rizos, "Birational solutions to the set-theoretical 4-simplex equation," to appear in Physica D: Nonl. Phen. (2023).
[2] S. Igonin, S. Konstantinou-Rizos,  "Set-theoretical solutions to the Zamolodchikov tetrahedron equation on groups and their Lax representations," arXiv:2302.03059 (2023)
[3] S. Konstantinou-Rizos, Noncommutative solutions to Zamolodchikov’s tetrahedron equation and matrix six-factorisation problems, Physica D: Nonl. Phen. 440 (2022), 133466.

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

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