The notion of multi-dimensional consistency has proven to be central in both the algebraic and geometric theory of discrete integrable systems. In this talk, we present a natural differential-geometric interpretation of a highly symmetric 4+4-dimensional dispersionless integrable differential equation. The multi-dimensional consistency of this (TED) equation arises naturally in the geometric context and, in fact, guarantees the consistency of the geometric construction leading to the underlying system of compatible TED equations. Emphasis is put on algebraic aspects so that those who are less familiar with the necessary geometric concepts may still be able to follow the discussion.

We explore how a tree on n vertices, through an associated weighted complete digraph, relates to a superintegrable n-component 3n-2 parameter family of Lotka-Volterra systems. These systems are measure-preserving, as is their Kahan discretisation.
Preprints:
Van der Kamp, Quispel and McLaren, Trees and superintegrable Lotka-Volterra families, arXiv …
Van der Kamp, McLachlan, McLaren and Quispel, Measure preservation and integrals for Lotka–Volterra T-systems and their Kahan discretisation, arXiv:2309.05979 [math.DS]

The faithful discretization of differential equations is an enduring topic. In this talk we give a (biased) survey of some of the novel discretization methods for ODEs, and the properties they preserve. We will also present some very recent new results in this area.
A decent introduction to this field is:
Petrera, Pfadler and Suris, On integrability of Hirota-Kimura discretizations, Reg & Chaotic Dyn 16 (2011) 245-289.
Collaborators: R McLachlan, D McLaren, P van der Kamp

We reexamined different examples of reduction chains g⊃g' of Lie algebras in order to show how the polynomials determining the commutant with respect to the subalgebra g' leads to polynomial deformations of Lie algebras. These polynomial algebras have already been observed in various contexts, such as in the framework of superintegrable systems. Two relevant chains extensively studied in Nuclear Physics, namely the Elliott chain su(3)⊃so(3) and the chain so(5)⊃su(2)×u(1) related to the Seniority model, are analyzed in detail from this perspective. We show that these two chains both lead to three-generator cubic polynomial algebras, a result that paves the way for a more systematic investigation of nuclear models in relation to polynomial structures arising from reduction chains. In order to show that the procedure is not restricted to semisimple algebras, we also study the chain Ŝ(3) ⊃ sl(2,R) × so(2) involving the centrally-extended Schrödinger algebra in (3+1)-dimensional space-time. This also allow to make connection between hidden symmetry and symmetry algebra and also lead to deformations of those models in context of reduction chains.
This is joint work with Rutwig Campoamor-Stursberg, Danilo Latini and Yao-Zhong Zhang. (arXiv:2303.00975)

I will give a pedagogical review of how integrable-models techniques can be used to perform exact computations in string theory, by treating the string worldsheet model as a two-dimensional quantum field theory in finite volume. I will also describe the significance of these results in unraveling the holographic``AdS/CFT'' correspondence, one of the major developments in quantum gravity of the last decades.

I will present a deformation of the 2d Toda hierarchy inspired by a correspondence with (refined) topological strings. It is derived by enhancing the underlying gl(∞) symmetry algebra to the quantum toroidal gl(1) algebra. The difference-differential equations of the deformed hierarchy are obtained from the expansion of (q,t)-bilinear identities, and two equations refining the 2d Toda equation are found in this way. I will also present an interesting class of solutions built from the R-matrix of the toroidal algebra.

In this talk, I will explain how computing the distributions of singularities of Painlevé functions is equivalent to solving inverse monodromy problems for Heun equations. This equivalence allows for the exact and asymptotic study of singularity distributions through application of Nevanlinna's theory of branched coverings of the Riemann sphere and complex WKB theory to Heun equations. As a main example, I will describe how this framework can be applied to the study of Wronskians of consecutive Hermite polynomials, yielding a proof of a conjecture by Peter Clarkson (2003).

Elliptic orthogonal polynomials are a family of special functions that satisfy certain orthogonality condition with respect to a weight function on an elliptic curve. Building up on several recent works on the topic, we establish a framework using Riemann-Hilbert problems to study such polynomials. When the weight function is constant, these polynomials relate to the elliptic form of the sixth Painlevé equation. This talk is based on a recent work with Tomas Latimer and Pieter Roffelsen (arXiv: 2305.04404).

The Riemann-Hilbert correspondence is a powerful tool to study the Painlevé differential equations. We study Birkhoff’s version on q-analogue of the Riemann-Hilbert problem and the Riemann-Hilbert-Birkhoff correspondence is also important to study the q-Painlevé equation. We mainly study the sixth q-Painlevé equation and we may also study the fifth q-Painlevé equation.

In this talk I will present formulations of consistency for 5-point lattice equations, particularly in lattices of type D and honeycomb lattices. The consistency property may be used to derive Lax matrices for 5-point equations. Known examples of consistent equations in both types of lattices give quadratic growths of degrees of iterates, indicating they are integrable.

We study the Stieltjes continued fraction expansion of a certain rational function of the plane on a hyperelliptic curve of genus g. We show how it gives rise to a birational map in dimension 3g+1 which has 2g+1 first integrals in involution with respect to a Poisson bracket. That is, we have a Liouville integrable map but more can be said: the map is actually algebraic completely integrable for each g. We christen them Volterra maps because they also provide genus g solutions of the infinite Volterra lattice equation. A particular case of the Volterra map (g=2) was previously found by Gubbiotti et al in a systematic search for 4D integrable maps.
This is joint work with Andy Hone (Kent) and Pol Vanhaecke (Poitiers).
See also https://arxiv.org/pdf/2309.02336.pdf

Treating the lattice sine-Gordon equation, along with its two simplest generalized symmetries, as a compatible system allows one to investigate another integrable system not directly connected to the sine-Gordon equation: a modified non-linear Schrödinger system with derivative.
The first step involves eliminating shifts from the two symmetries of the lattice sine-Gordon equation, resulting in the NLS-type system. Subsequently, an auto-Bäcklund transformation and a superposition formula for the NLS-type system are obtained by eliminating shifts from both the lattice sine-Gordon equation and its down-shifted version. We employ the derived formulas to calculate a superposition of two and three elementary solutions.

I will first introduce two types of random matrices and discuss classical results such as the semicircle law and the Tracy-Widom law. Then I will present our recent findings regarding extreme gap problems in classical random matrices and propose several conjectures.

A constant mean curvature (cmc) torus in R³ can be described in terms of a hyperelliptic spectral curve, whose genus g is at least two. The countably infinite family of examples studied by Wente and Abresch all have g=2, but there are many more cmc tori in this simplest class. I shall parameterise the closure of spectral data of these constant mean curvature tori in R³ as an isoceles right-angled triangle, with the Wente family along its diagonal. Furthermore, the boundary of this triangle contains solutions to a number of different integrable systems: sinh-Gordon, NLS, KdV and non-conformal harmonic maps all make an appearance.

Consider a simple mechanical system proposed by Boltzmann in the 1860's: a massive particle moves in a gravitational field with a linear boundary between the particle and the center of gravity. Reflections off the boundary are elastic and obey the billiard reflection law: angles of incidence and reflection are congruent. This system was recently shown by Gallavotti and Jauslin to have a second integral of motion. We study its dynamics and prove the existence of caustics, Cayley-type periodicity conditions, and more. This is joint work with Milena Radnović (University of Sydney).