I will talk about  continuous limits, normal forms, possibly symmetries, etc. The course is suitable for 2d year students in general or above.

The lecture will be in English.

Venue: Room 419, 7th corpus, YarSU.

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: Among the invariant characteristics of dynamical systems, an important role is played by the Lyapunov exponents and the Lyapunov dimension. The analysis of the spectrum of Lyapunov exponents is widely used to study complex dynamics in systems of ordinary differential equations and in models that reduce to mappings. In the finite-dimensional case, according to the Oseledets theorem, the system of ordinary differential equations linearized on an attractor is always Lyapunov stable, and thus the upper limit can be replaced by the ordinary one, which makes it possible to effectively calculate the Lyapunov exponents. In this talk, it is planned to consider the question of calculating Lyapunov exponents for systems of differential equations of argument with delay, for which the theorem in general does not work. The results of testing the developed algorithm for the Hutchinson equation will be presented and an application of the algorithm to some problems will be illustrated.

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

Abstract: TBA.

Location: Delone laboratory of "Discrete and Computational Geometry", 3 Komsomolskaya rd.

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: TBA.

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

We invite everyone to attend the lectures of Prof. V.V. Sokolov, Head Researcher of the Landau Institute for Theoretical Physics of the Russian Academy of Sciences.

The course of lectures is aimed at 1-st and 2-nd year students, but everyone is welcome.

In total, 2 meetings are planned:
The first meeting: Tuesday May 15, 16:00-17:30, lect. theatre 309, 7th building of YSU
The second meeting: Thursday May 17, 16:00-17:30, lect. theatre 309, 7th building of YSU

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)