## Second International Conference "Integrable Systems and Nonlinear Dynamics 2020"

The conference will be held in the city of **Yaroslavl, Russia, 5-10 October 2020.**

The conference will be held in the city of **Yaroslavl, Russia, 5-10 October 2020.**

**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

Sergei Igonin will give lectures on "English for mathematicians" for everyone.

The lectures will be taking place on **Thursdays** (**November 29**, **December 6**, **December 13**) from **14:15** to **15:45**.

The lectures are also planned to be continued in the next semester.

**Venue:** Lecture theatre 317, 7th building YarSU (144 Soyuznaya str.).

Learning materials can be found on the website.

https://vk.com/ma3333

**Abstract:**

Discussion of the following topics is planned.**1.** Effective ways to learn English.**2.** Features of the use of English in communicating about mathematics.**3. **How to make reports and write mathematical texts in English.

Materials from leading universities in Moscow and the UK will be used.

The lecturer has working experience in English at the University of Leeds (United Kingdom).

Boris Bychkov (Higher School of Economics) will give a lecture on "Symmetric polynomials" for 1st and 2nd year students and school students.

**The lecture will take place on November 21, 2018, at 17:30.**

**Venue: **8/10 Kirova st. (2nd building of YarSU), aud. 202.

**Abstract: **A polynomial in several variables is called symmetric if it is invariant with respect to any permutations of variables. The main theorem on symmetric polynomials asserts that any symmetric polynomial can be expressed in terms of elementary ones, and in a unique way. We shall obtain several more such sets of polynomials, and then we shall define Schur polynomials, a basis in the space of symmetric polynomials, parametrised by partitions (Young diagrams), and will discuss some interesting properties of this basis.

Vladimir Nikolaevich Rubtsov from the University of Angers will give lectures for first and second year students and school students.

**1nd and 2nd lecture:** November 22, 2018, at 16:00

**3rd and 4th lecture:** November 23, 2018, at 16:00

**Venue: **2nd corpus, YarSU, aud. 106.

**Mini-course title:** "Complex numbers, their generalisations, geometry and applications."**Abstract:** In my lectures, I will talk about various extensions of natural and real numbers, the basic algebraic structures and operations associated with these extensions, as well as the conditions of complexity and their applications.

2nd lecture on this topic.

**Abstract:** 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: **Laboratory 419, 7-th corpus, YarSU.

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.