Lectures and topics

From 3D QRT to discrete Painlevè equations by Jaume Alonso (Technische Universität Berlin Berlin)

In this cycle of talks, we introduce a generalisation of the Quispel–Roberts–Thompson (QRT) construction to 3D, we explain its applications to the Kahan-Hirota-Kimura (KHK) discretisation and we illustrate how it can be related to the discrete Painlevé equations. QRT maps in 2D are a composition of two involutions defined using a pencil of biquadratic curves in ℙ¹ × ℙ¹. Its generalisation to 3D is likewise a composition of two involutions but defined using two pencils of quadrics, resulting in an integrable birational map in ℙ³ of degree 25. We show how, under certain circumstances, these maps can become of degree 5 and even 3. Several maps of this kind exist already in the literature, for example, the KHK discretisations of the Euler top and the Zhukovski–Volterra gyrostat with one linear parameter. As it happened with the 2D case, the 3D QRT construction can also be used to “correct” KHK discretisations, i.e. take a non-integrable discretisation and make it integrable. The last lecture describes the transformation process by which a 3D QRT map can be deformed into discrete Painlevé equations. The geometry in terms of pencils of quadrics offers an explanation of the de-autonomisation
required to obtain discrete Painlevé equations.

r-matrices for integrable systems by Marta Dell’Atti (University of Warsaw)

We consider some algebraic and geometric aspects of the theory of integrable systems in finite dimensions, associated with the existence of a classical r-matrix, first introduced by Sklyanin as the classical analogue of the quantum version. The importance of the notion of the r-matrix in this context relies on the fact that it connects the Hamiltonian structure of integrable equations with the factorisation problem which provides their explicit solution.

In this framework, the Lax matrix is interpreted as the coadjoint orbit of a Lie algebra 𝔤, and the existence of a non-dynamical r-matrix
allows the introduction of a second Lie algebra structure on 𝔤. Depending on the properties of the r-matrix associated with the specific
system, we distinguish between bi-algebras and di-algebras. The first are associated with a skew-symmetric r-matrix and were introduced by Drinfeld, the second refer to a larger class of r-matrix and are related to the factorisation properties of the system. A particular class is given by the factorisable Lie bi-algebras, whose Poisson bracket takes the form of the Sklyanin brackets.

We will introduce the classical and modified Yang-Baxter equations, Poisson manifolds, Lie-Poisson brackets and Poisson-Lie groups and apply the formalism to construct some dynamical systems in finite dimension.

The role of integrability in Fermi-Pasta-Ulam-Tsingou-like models by Matteo Gallone (SISSA)

Seventy years ago, in 1955, Enrico Fermi, John Pasta, Stanislaw Ulam and Mary Tsingou (FPUT) published a report on a series of numerical experiments performed using the computer Maniac II in Los Alamos. Their aim was to understand heuristically the dynamics of certain one-dimensional chains of particles, especially in relation with fundamental problems in statistical mechanics and hydrodynamics. Their initial expectations were completely destroyed by the outcome of the numerical experiments: instead of showing a
slow relaxation to thermal equilibrium, numerical experiments displayed an unexpected recurrent dynamics. This was part of the motivations that led Zabusky and Kruskal study the Korteweg-de Vries equation (KdV) and to guess that the unexepcted outcome of the FPUT experiment was due to an integrable structure of the KdV, that has later been discovered.

Nowadays, lack of quick thermalization can often be described by means of vicinity of integrable models and such phenomena can be observed in laboratory experiments due to our technological improvements in quantum technologies. In these lectures, I will
introduce the FPUT model, its motivations and discuss the most advanced results available in the literature and, in particular, its relation with KdV hierarchy and the Toda chain.

Classical algebraic geometry and integrable systems by Michele Graffeo (SISSA) and Alexander Stokes (Waseda University)

Classical algebraic geometry interacts with the theory of integrable systems in many ways. For instance, the theory of elliptic surfaces, or more generally of generalised Halphen surfaces, play a preeminent rôle in the Okamoto–Sakai theory of differential and difference Painlevé equations. Analogously, the notion of singularity confinement and algebraic entropy are deeply related to the general properties of birational maps of the complex projective space. So, the aim of these lectures is to present an hands-on introduction to the tools from classical algebraic geometry that are needed to face problems coming from integrable systems theory.

In the first part of the lectures, we will discuss with explicit formulas and examples the resolution of singularities of maps and surfaces through the blow up procedure and its generalisations, the calculation of the action on cohomology of a map, and symplectic structures on algebraic varieties.

In the second part of the lectures we will see some example, mainly in dimension two and three, of the theory in action. That is, we will construct what is known as the space of initial conditions for some differential and difference equations, and show in some selected cases the exact calculation of the algebraic entropy. Finally, a shortndiscussion of Hamiltonian structures and symplectic atlases will be given.

Hamiltonian operators for integrable differential-difference equations by Edoardo Peroni (University of Kent)

Integrability is a concept that is challenging to define in a unique way: there are many aspects of it, not all of which are always present. One common feature is the existence of different Hamiltonian structures, and this characteristic is not exclusive to partial differential equations (PDEs). In these notes, we are considering the formalism of Hamiltonian operators applied to integrable
differential-difference equations (DΔEs) as integrable systems dependingon a discrete variable.

In the first part, we introduce the necessary formalism for understanding DΔEs, such as difference field and difference operators. This foundation allows, in the second section, a self-contained definition of Hamiltonian operators on the difference field. We then apply this framework to the study of integrable DΔEs. Additionally, we cover concepts such as pre-Hamiltonian, Nijenhuis and recursion operators.

When discussing the Hamiltonian structure of DΔEs, it is helpful to draw parallels with the analogous concepts from the continuous case. Throughout the notes, we emphasize these connections by revisiting key definitions within the context of PDEs. Additionally, exercises and examples are included to deepen understanding and explore the material’s implications in more detail.

Orthogonal polynomials and discrete Painlevé equations: theory and applications by Sofia Tarricone (Sorbonne Université)

In these lectures we will study how discrete Painlevé equations appear in the theory of orthogonal polynomials. The occurrence of this phenomenon is due to the fact that families of orthogonal polynomials can be described via recurrence relations, either the three-terms recurrence relation in the case of orthogonality measure on the real line or the Szego recurrence relation in the case of orthogonality measure on a general curve of the complex plane. As it turned out, the recurrence coefficients appearing in these relations satisfy, for
certain type of measures, nonlinear discrete equations of discrete Painlevé type with coefficients depending on the parameters of the measure.

In the first part of the lectures, we will review the classical theory of orthogonal polynomials on the real line (OPRL for short) introducing
as well the Riemann-Hilbert approach. We will then focus on the particular case of orthogonality measure being a deformation of the Gaussian measure on the real line, to see how the discrete Painlevé I equation arises in this setting. In the second part of the lectures, we will see how the theory of OPRL can be extended to the case of orthogonal polynomials with a measure on the unit circle in the complex plane (OPUC for short). We will then focus on a specific orthogonality measure to see how this time, the discrete Painlevé II equation describes the recurrence satisfied by the Versblusnky coefficients of this family of orthogonal polynomials.

Finally, we will see some applications of these results, respectively in matrix models appearing in 2D quantum gravity and in the Ulam problem for random permutations. In particular, we will see how the study of continuous limit of the discrete Painlevé I and II equations appearing in these contexts, via their relation with orthogonal polynomials, led to fundamental asymptotic results.