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.

Abstract: We have found a quite general construction of commuting vector fields on the $k$-th symmetric power of $\mathbb{C}^{m}$ and tangent vector fields to the $k$-th symmetric power of an affine variety $V\subset\mathbb{C}^{m}$. Application of this construction to the $k$-th symmetric power of a plane algebraic curve $V_g$ of genus $g$ leads to $k$ integrable Hamiltonian systems on $\mathbb{C}^{2k}$ (or on $\mathbb{R}^{2k}$, if the base field is $\mathbb{R}$). In the case $k=g$, the symmetric power ${\rm Sym}^k(V_g)$ is birationally isomorphic to the Jacobian of the curve $V_g$, and our system is equivalent to well-known Dubrovin's system, which was derived and studied in the theory of finite-gap solutions (algebro-geometric integration) of the Korteweg–de Vries equation. We have found coordinates in which the obtained systems and their Hamiltonians are polynomial. For $k=2,\ g=1,2,3$ we present these systems explicitly and discuss the problem of their integration.

Venue: 7th corpus YarSU, lecture theatre 419

Abstract: Integrability in statistical physics models usually means that the partition function can be represented in terms of the transfer matrix included in the "large" commutative family. The last property for two-dimensional models is traditionally accompanied by the structure of a vertex model with a weight matrix satisfying the Yang-Baxter equation. This talk is about the generalisation of this idea to a larger dimension, in particular, I will consider the three-dimensional Ising model, as well as the Hopfield neural network model on a 2-dimensional triangular lattice in the memory phase. It turns out that both these models have a vertex representation, with a matrix of weights that satisfies the deformation of the generalization of the Yang-Baxter equation in 3 dimensions, the so-called twisted equation of tetrahedra. In both cases, the combinatorics of the hypercube is essentially used to construct the matrix of weights.

Location: 3 Komsomol'skaya st., B. Delaunay "Discrete and computational geometry" laboratory.

Abstract: All Painlevé equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic). Recently, these systems of interacting particles have been proved to be relevant the study of β-models.

An almost two decade old open question by Takasaki asks whether these multi-particle systems can be understood as isomonodromic equations, thus extending the Painlevé correspondence. I shall give an (afrmative ) answer by displaying explicitly suitable isomonodromic Lax pair formulations. As an application of the isomonodromic representation we provide a construction based on discrete Schlesinger transforms, to produce solutions for these systems for special values of the coupling constants, starting from uncoupled ones; the method is illustrated for the case of the second Painlevé equation.

This is a joint work with Marco Bertola (SISSA-CRM, Montreal) and Mattia Cafasso (LAREMA, Angers).

References

[1] M. Bertola, M. Cafasso, V. Roubtsov, Non-commutative Painlevé equations and systems of Calogero type, arXiv:1710.00736, 25 pp.

[2] K. Takasaki. Painlevé-Calogero correspondence revisited. J. Math. Phys., 42(3):1443–1473, 2001.

Location: 3 Komsomol'skaya st., B. Delaunay "Discrete and computational geometry" laboratory.

Abstract: Many interesting examples of discrete integrable systems can be studied from the geometric point of view. In this talk we will consider two classes of examples of such system: autonomous (QRT maps) and non-autonomous (discrete Painlevé equations). We introduce some geometric tools to study such systems, such as the blowup procedure to construct algebraic surfaces on which the mappings are regularized, linearization of the mapping on the Picard lattice of the surface and, for discrete Painlevé equations, the decomposition of the Picard lattice into complementary pairs of the surface and symmetry sub-lattices and construction of a birational representation of affine Weyl symmetry groups that gives a complete algebraic description of our non-linear dynamic. If time permits, we also explain the relationship between this picture and classical differential Painlevé equations.

Location: Znam. tower, 3 Komsomol'skaya rd. (entrance inside the arch).

Abstract: By a quantum matrix algebra (QMA) I mean an algebra generated by the entries of a matrix subject to some relations. The best known QMAs are the RTT algebras and the Reflection Equation ones. Each of them is associated with an R-matrix (constant or depending on parameters). QMAs play a very important role in the Quantum Integrable System theory. I plan to exhibit some properties of the mentioned algebras and explain their role in this theory.

Location: Znamenskaya Tower, 3 Komsomolskaya St. (entrance inside the arch)