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

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