Speaker:  Frank Nijhoff (University of Leeds, UK)

Date and time:  05.05.2021, 17:00 Moscow time (GMT +03:00) / 15:00 UK time

Title: Elliptic integrable systems on the space-time lattice

Abstract:  I will review the status of various integrable lattice equations (partial difference equations) of elliptic type, together with their integrability aspects, associated Lax systems, explicit solutions and higher-rank generalisations.

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

Speaker:  Vladimir Sokolov (Landau Institute for Theoretical Physics, Chernogolovka, Russia and UFABC, Sao Paulo, Brazil)

Date and time:  21.04.2021, 17:00 (GMT +03:00)

Title: Non-Abelian generalizations of integrable PDEs and ODEs

Abstract:  A general procedure for non-abelinization of а given integrable polynomial differential equation is described. We are considering NLS type equations as an example. We also find non-abelinating Euler's top. Results related to the Painlevé-2 and Painlevé-4 equations are discussed.

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

Speaker:  Pavlos Xenitidis (Liverpool Hope University, UK)

Date and time:  07.04.2021, 17:00 (GMT +03:00)

Title: Symmetries and Integrability of Difference Equations

Abstract:  Symmetries provide arguably the most reliable means to test and prove the integrability of a given equation. They are used in the analysis and classification of partial differential equations and differential-difference equations since 1970’s and 1980’s, and only very recently this approach has been extended to partial difference equations. In this talk I will consider a class of partial difference equations in two dimensions and discuss the general form of their symmetries. I will derive necessary integrability conditions for these equations and explain how they lead to symmetries and conservation laws. I will also demonstrate how symmetries can be used to find solutions and reduce a partial difference equation to discrete Painlevé type equations. Finally, I will discuss several extensions of the theory to other classes of scalar equations and to systems of difference equations.

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

Speaker:  Anton Dzhamay (University of Northern Colorado)

Date and time:  24.03.2021, 17:00 (GMT +03:00)

Title: Discrete Schlesinger equations and difference Painlevé equations

Abstract:  The theory of Schlesinger equations describing isomonodromic dynamic on the space of matrix coefficients of a Fuchsian system w.r.t. continuous deformations is well-know. In this talk we consider a discrete version of this theory. Discrete analogues of Schlesinger deformations are Schlesinger transformations that shift the eigenvalues of the coefficient matrices by integers. By discrete Schlesinger equations we mean the evolution equations on the matrix coefficients describing such transformations. We derive these equations, show how they can be split into the evolution equations on the space of eigenvectors of the coefficient matrices, and explain how to write the latter equations in the discrete Hamiltonian form. We also consider some reductions of those equations to the difference Painlevé equations, again in complete parallel to the differential case. 

This is a joint work with H. Sakai (the University of Tokyo) and T. Takenawa (Tokyo Institute of Marine Science and Technology).

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

Speaker:  Andrei Vesnin (Tomsk State University)

Date and time: 24.02.2021, 17:00 (GMT +03:00)

Title: Hyperbolic polyhedra and hyperbolic knots: the right-angled case

Abstract:  A polyhedron is said to be right-angled if all its dihedral angles are equal to pi/2. Three-dimensional hyperbolic manifolds constructed from bounded right-angled polyhedra have many interesting properties [1]. Inoue [2,3] initiated enumerating bounded right-angled hyperbolic polyhedra by their volumes. Atkinson obtained low and upper bounds of volumes via vertex number [4]. In [5] we enumerate ideal (with all vertices at infinity) right-angled hyperbolic polyhedra. The obtained results imply that the right-angled knot conjecture from [6] holds for knots with small crossing number. Atkinson’s upper bounds were improved in [7,8] for bounded and ideal cases both. Finally, we will discuss the relation of results from [5] with the maximum volume theorem from [9].

The talk is based on joint results with Andrey Egorov [5,7,8].

References.

[1] A. Vesnin, Right-angled polyhedra and hyperbolic 3-manifolds, Russian Mathematical Surveys 72 (2017), 335-374.

[2] T. Inoue, Organizing volumes of right-angled hyperbolic polyhedra, Algebr. Geom. Topol. 8 (2008), 1523-1565.

[3] T. Inoue, Exploring the list the smallest right-angled hyperbolic polyhedra, Experimental Mathematics 2019, published online.

[4] C. Atkinson, Volume estimates for equiangular hyperbolic Coxeter polyhedra, Algebr. Geom. Topol. 9 (2009), 1225-1254.

[5] A. Vesnin, A. Egorov, Ideal right-angled polyhedra in Lobachevsky space, Chebyshevskii Sbornik 21 (2020), 65-83.

[6] A. Champanerkar, I. Kofman, J. Purcell, Right-angled polyhedra and alternating links, arXiv:1910.13131.

[7] A. Egorov, A. Vesnin, Volume estimates for right-angled hyperbolic polyhedra, Rendiconti dell’Instituto di Matematica dell’Universita di Trieste 52 (2020), 565-576.

[8] A. Egorov, A. Vesnin, On correlation of hyperbolic volumes of fullerenes with their properties, Comput.  Math. Biophys. 8 (2020), 150-167.

[9] G. Belletti, The maximum volume of hyperbolic polyhedral, Trans. Amer. Math. Soc. 374 (2021), 1125-1153.

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

Speaker:  Igor G. Korepanov

Date and time: 23.12.2020, 17:00 (GMT +03:00)

Title: Functional tetrahedron equation and related structures

Abstract: We show how the functional tetrahedron equation appears, together with its solutions, from almost nothing -- lines in a plane and simple linear algebra. Then, a discrete-time dynamical system arises naturally, with an algebraic curve conserved in time, and divisors moving linearly along their Picard group. Finally, a reduction of this system is constructed, for which the conserved quantities turn into the partition function of an inhomogeneous six-vertex free-fermion model.

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

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

Date and time: 9.12.2020, 17:00 (GMT +03:00)

Title: Quantum uniformisation and CY algebras

Abstract: In this talk, I will discuss a special class of quantum del Pezzo surfaces. In particular I will introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations. I will try to explain (at least a part of) terminology above.

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

Speaker: Anton Izosimov (University of Arizona, USA)

Date and time: 6.05.2020, 18.00 (Moscow time)

Title:  The pentagram map, Poncelet polygons and commuting difference operators

Abstract: The pentagram map is a discrete integrable system on the space of projective equivalence classes of planar polygons. By definition, the image of a polygon P under the pentagram map is the polygon P' whose vertices are the intersection points of consecutive shortest diagonals of P, i.e. diagonals connecting second nearest vertices. In the talk, I will discuss the problem of describing the fixed points of the pentagram map. In other words, the question is: which polygons P are projectively equivalent to their "diagonal" polygons P'? It is a classical result of Clebsch that all pentagons have this property. Furthermore, in 2005 R.Schwartz proved that this property is also enjoyed by all Poncelet polygons, i.e. polygons that are inscribed in a conic section and circumscribed about another conic section. In the talk I will argue that in the convex case the converse is also true: if P is convex and projectively equivalent to its diagonal polygon P', then P is a Poncelet polygon. The proof is based on properties of commuting difference operators, real elliptic curves, and theta functions.

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

Lecturer: Sergei Smirnov, MSU
Talk title: Discretisation of two-dimensional Toda Chains 
Time: February 20, at 16:00 
Venue: 7th building YarSU, seminar room 427 (144 Soyuznaya str., 150007)

Abstract:
It is well known that, in the continuous case, the two-dimensional Toda chains corresponding to the Cartan matrices of simple Lie algebras are Darboux integrable, that is, integrable in an explicit form, and the chains corresponding to the generalised Cartan matrices are integrable by the inverse scattering method. 

Although discrete versions of particular cases were considered earlier, in 2011 I.T. Khabibullin proposed a systematic method for the discretisation of the so-called exponential type systems (generalisation of Toda chains): the idea was to find a discretisation in which the characteristic integrals during the transition from the continuous to the semi-discrete model (and from the semi-discrete to the purely discrete) retain their form. The articles by Khabibullin et al. demonstrated that this method works for Toda chains of small length.

I will explain why this method works for discretising chains of arbitrary lengths of the series A and C and what is the progress in the question of the integrability of these discretisations in the general case.