Speaker:  Eugene Rogozinnikov (Max-Planck-Institut für Mathematik in den Wissenschaften (MPI MiS), Leipzig, Germany)

Date and time:  10.04.2024, 17:00 (GMT +03:00)

Title: Hermitian Lie groups as symplectic groups over noncommutative algebras

Abstract: In my talk (based on a joint work with D. Alessandrini, A. Berenstein, V. Retakh and A. Wienhard), I introduce the symplectic group $Sp_2(A,\sigma)$ over a noncommutative algebra $A$ with an anti-involution $\sigma$ and show that many Lie groups can be seen in this way. Of particular interest will be the classical Hermitian Lie groups such as $Sp(2n,R)$, $U(n,n)$ and their complexifications. For these groups, I realize their symmetric space in terms of $Sp_2(A,\sigma)$ thus generalizing several famous models of the hyperbolic plane and the three-dimensional hyperbolic space. Our construction has a flavor of noncommutative projective line over the complexification of $A$ which is always a compact symmetric space when $A$ Hermitian and semisimple or its complexification. We expect it to hold for any semisimple $A$. This, in turn, would imply that $Sp_2(A,\sigma)$ is reductive when $A$ is semisimple.

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

Speaker:  Ismagil Habibullin (Ufa Federal Research Centre of Russian Academy of Science)

Date and time:  27.03.2024, 17:00 (GMT +03:00)

Title: On the classification of nonlinear integrable chains in 3D

Abstract: Nonlinear integrable equations with three independent variables, at least one of which is discrete, may admit boundary conditions that include discontinuities in the discrete variable while still maintaining the integrability of the equation. This means that by imposing these boundary conditions on the two ends of a segment, one can obtain an integrable system of two-variable equations. Integrable systems have a broad range of boundary conditions that allow for discontinuities and still maintain the integrability properties of the system. In some cases, these discontinuity conditions result in systems that are soliton-like, and it has been shown that there exists a single special discontinuity condition that yields an integrable finite system in the Darboux sense. In our recent work, we have demonstrated that this type of special reduction, which can be of arbitrary order, can be successfully applied to solve the problem of classifying three-dimensional integrable systems.

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

Speaker:  Vladimir Retakh (Rutgers University)

Date and time:  13.03.2024, 17:00 (GMT +03:00)

Title:  The noncommutative Laurent phenomenon

Abstract: The composition of polynomials is always a polynomial, but this is not the case for Laurent polynomials containing inverse variables. When Laurent property persists, one talks about the Laurent phenomenon. In the commutative case, the Laurent phenomenon is closely related to the theory of cluster algebras. I will talk about examples of the noncommutative Laurent phenomenon and its connection with the theory of noncommutative cluster algebras. The report is based on joint work with Arkady Berenstein.

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

Speaker:  Vincent Caudrelier (School of Mathematics, University of Leeds)

Date and time:  28.02.2024, 17:00 (GMT +03:00)

Title: Soliton interactions, Yang-Baxter and reflection maps and their Poisson properties

Abstract: Using the vector nonlinear Schrödinger equation as the main example, I will briefly review how certain solutions of the set-theoretical Yang-Baxter equation, called Yang-Baxter maps, arise from the interactions of multicomponent solitons. This is best seen using the Zakharov-Shabat dressing method and refactorisation of the elementary dressing factors. In this purely classical context, it is remarkable that the Yang-Baxter equation also ensures that the total scattering map describing the collisions consistently factorises into a product of two-soliton collisions, just like in the more well-known quantum context. I will then discuss the problem of integrable boundary conditions and explain how it leads to the introduction of the set-theoretical reflection equation. Solutions to this equation, called reflection maps, arise from the reflection of a soliton on the boundary. Again, the complete analogy between this context and the more well-known quantum reflection equation introduced by Cherednik and Sklyanin holds. Finally, I will present results on the symplectic and Poisson properties of these maps. This is a natural problem to consider given the interpretation (reviewed e.g. in Faddeev-Takhtajan's book) of soliton dynamics in the scalar case as canonical transformations.

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

Speaker:  Alexandra Garazha (MSU, Moscow)

Date and time:  31.01.2024, 17:00 (GMT +03:00)

Title: Kronecker's method and complete systems of functions in bi-involution on classical Lie algebras

Abstract: We examine the relation of the integrability property and the bi-Hamiltonian structure, the latter means that there are two compatible Poisson brackets and the dynamics is Hamiltonian with respect to both of them. We study the simplest case when one bracket is linear and the other one is constant. Namely, we will consider a classical simple Lie algebra g and define two Poisson brackets: the classical Lie-Poisson bracket and the bracket "with frozen argument A", which can be constructed for any A ∈ g. 

If A is regular, the corresponding complete systems of functions in bi-involution can be obtained by the Mishchenko-Fomenko argument shift method. We will show how to generalize this method to the case of an arbitrary element A using an algebraic approach.

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

Speaker:  Vladimir Roubtsov (Universit'e d'Angers, LAREMA, ITTP RAS)

Date and time:  13.12.2023, 17:00 (GMT +03:00)

Title: Kontsevich and Buchstaber polynomials, multiplication kernels and differential operators of the Calabi-Yau type

Abstract: We are discussing several recent results of ongoing work (in collaboration with I. Gayur and D. By Van Straten and with V. Buchstaber and I. Gayur) about interesting properties of multiplicative generalized Bessel kernels, which include the well-known Clausen and Sonin-Gegenbauer formulas, examples of Kontsevich discriminant locus polynomials given as addition laws for special two-valued formal groups (Buchstaber-Novikov-Veselov), as well as the connection with "period functions" solving some Picard-Fuchs-type equations for the Calabi-Yau cases and related to analogues of Landau-Ginzburg superpotentials.

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

Speaker:  A. Sleptsov (NRC ”Kurchatov Institute”, MIPT, IITP)

Date and time:  29.11.2023, 17:00 (GMT +03:00)

Title: Tug-the-hook symmetry for quantum 6j-symbols

Abstract: Quantum 6j-symbols (Racah-Wigner coefficients) provide isomorphism between two different fusions in tensor product of given three representations of a quntum group. They appear in many contexts: theory of angular momenta in quantum mechanics, calculation of amplitudes in 3d quantum gravity models. In my talk, I will first define the 6j-symbols for the quantum group U_q(sl_N) and review their most important properties: tetrahedral symmetry, the pentagon relation (the Biedenharn-Elliot identity), connections to hypergeometric functions and explicit formulas. Then I’ll talk a little about relations of 6j-symbols with quantum invariants of knots, Turaev-Viro invariants for 3-manifolds, and correlators in (conformal) field theory. Finally, I'll talk about our tug-the-hook conjecture, which suggests the existence of a new large class of symmetries for 6j-symbols.

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

Speaker:  Ivan Kozlov

Date and time:  15.11.2023, 17:00 (GMT +03:00)

Title: Jordan-Kronecker invariants and Integrable Hamiltonian systems

Abstract: The Jordan-Kronecker invariants of a complex finite-dimensional Lie algebra were introduced by A.V. Bolsinov and P. Zhang in [1]. In short, they are the canonical form of a pencil of skew-symmetric bilinear forms.

In the talk we will discuss the latest results about the Jordan-Kronecker invariants and their relation with the integrability of Hamiltonian systems. In particular, we will talk about the new results from [2] and [3].

[1] A.V. Bolsinov and P. Zhang “Jordan-Kronecker invariants of finite-dimensional Lie algebras”, Transformation Groups, 21:1 (2016), 51–86

[2] I.K. Kozlov “Realization of Jordan-Kronecker invariants by Lie algebras”, arXiv:2307.08642 [math.DG]

[3] I.K. Kozlov “Shifts of semi-invariants and complete commutative subalgebras in polynomial Poisson algebras”, arXiv:2307.10418 [math.RT]

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

Speaker: Dmitry Talalaev (MSU, YarSU, ITEP)

Date and time:  1.11.2023, 17:00 (GMT +03:00)

Title: Multivalued quandles and corack n-bialgebras

Abstract: Multivalued groups arise in a fairly wide field of mathematical subjects: representation theory, discrete dynamical systems, algebraic geometry. The report  is focused on a similar construction in the category of quandles, structures that have significant applications in knot theory and discrete integrable systems.

After short introduction to the theory of n-valued groups I will define the concept of an n-quandle, give some examples, describe the coset construction in this case, talk about a natural analogue of a group algebra for a quandle, and construct an n-corack bialgebra associated with a multivalued quandle.

The report is based on joint results with V. Bardakov, V. Buchstaber and T. Kozlovskaya.

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

Speaker: V. V. Sokolov (IITP RAS, Moscow)

Date and time:  18.10.2023, 17:00 (GMT +03:00)

Title: Non-Abelian Poisson brackets

Abstract: Poisson brackets on the vector space of traces of non-commutative polynomials in several matrix variables and their generalization to the case of free associative algebra are considered. A projectivization of such brackets is defined to construct non-Abelian elliptic Poisson brackets of the Sklyanin-Odesskii-Feigin type.

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