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

Speaker: Oleg K. Sheinman (Steklov Mathematical Institute, Moscow)

Date and time:  4.10.2023, 17:00 (GMT +03:00)

Title: Obstructions for Inverse Spectral Method and for Separation of Variables for Hitchin systems

Abstract: Inverse Spectral Method is a fundamental method with which the main progress of XX-XXIth centuries in the soliton theory is related. For the finite gap solutions, its algebraic-geometric version enables one to explicitly express such solutions via theta functions. It is also applicable to integrable systems with finitely many degrees of freedom, admitting a Lax representation. In particular, Krichever (2002) proposed a way to explicitly integrate Hitchin systems on the moduli space of all (semi-stable) holomorphic bundles. However, further investigations showed that there is a certain obstruction for Krichevers approach for the Hitchin systems on moduli spaces of G-bundles where G is a complex simple Lie group (which we refer to as the structure group of the system). This obstruction is related to the fact that dynamical poles of Baker-Akhieser functions are anavoidable in this case.

A similar phenomena can be observed for another fundamental method of the theory of integrable systems, namely for the Method of Separation of Variables. It gives theta function formulae for solutions of the GL(n) Hitchin systems but there appears an obstruction for the systems with simple structure groups. The last observation is related to the peculiarities of the inversion problem for Prim varieties.

In the talk, I shall define Hitchin systems via their Lax representation (due to I. Krichever), explain the Inverse Spectral Method for them, and give a quite simple calculation demonstrating the essence of the obstruction. Then I'll do the same for Separation of Variables.

Speaker: Alexander V. Mikhailov (University of Leeds)

Date and time:  13.09.2023, 17:00 (GMT +03:00)

Title: A novel approach to quantisation of dynamical systems

Abstract: We propose to revisit the problem of quantisation and look at it from an entirely new angle, focussing on quantisation of dynamical systems themself, rather than of their Poisson structures. We begin with a lift of a classical dynamical system to a system on a free associative algebra with non-commutative dynamical variables and reduce the problem of quantisation to the problem of studying of two-sided quantisation ideals, i.e. the ideals of the free algebra that define the commutation relations of the dynamical variables and are invariant with respect to the non-commutative dynamics. Quantum multiplication rules in the quotient algebra over a quantisation ideal are manifestly associative and consistent with the dynamics. We found first examples of bi-quantum systems which are quantum counterparts of bi-Hamiltonian systems in the classical theory. Moreover, the new approach enables us to define and present first examples of non-deformation quantisations of dynamical systems. The new approach also sheds light on the problem of operator's ordering.

Bibliography:

[1] A.V.Mikhailov, Quantisation ideals of nonabelian integrable systems.  Russ. Math. Surv., 75(5):199, 2020.

[2] V.M.Buchstaber and A.V.Mikhailov, KdV hierarchies and quantum Novikov's equations. arXiv:2109.06357.

[3] S.Carpentier,  A.V.Mikhailov and J.P.Wang. Quantisation of the Volterra hierarchy. Lett. Math. Phys., 112:94, 2022.

Speaker: Dmitry Talalaev (MSU, YarSU, ITEP)

Date and time:  7.06.2023, 17:00 (GMT +03:00)

Title: Polynomial graph invariants, Sturm criterion, hyperbolic stability and total positivity

Abstract: I will talk about various manifestations of the phenomenon of total positivity in such seemingly remote areas as polynomial invariants of graphs, problems of localization of the roots of a polynomial, the geometry of flag varieties. Nevertheless, in all these problems, the condition we are interested in is expressed in terms of total positivity, or n-positivity, of some auxiliary matrix: Toeplitz matrix, Hankel matrix, or Sturm matrix, depending on the problem.

In particular, I will talk on logarithmic concavity of the conditional flow and chromatic polynomial and the criterion of hyperbolic stability of a polynomial.

The report is based on several joint works with B. Bychkov, D. Golitsyn, A. Kazakov, A. Kutuzova, E. Lerner and S. Mukhamedzhanova.

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

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