We will show that the ODE describing the Lagrange top (possibly with an added harmonic potential term) with appropriately time-dependent moments of inertia and/or potential can be transformed into the Painlevé V equation. It is well known that the Painlevé equations are Hamiltonian with time-dependent Hamiltonians, but the link to the Lagrange top appears to be new. The connection appeared through the quantisation of the Lagrange top. The Schrodinger equation for the Lagrange top is a confluent limit of a Fuchsian equation, specifically the confluent Heun equation. Painlevé V can be thought of as a "de-quantisation" of the confluent Heun equation, and for the Lagrange top these statements can be made precise.

Recently, we reviewed [1] some properties of the affine Weyl group in the context of their applications to discrete integrable systems such as the discrete Painlevé equations [2]. In particular, a dual representation is used to discuss translational elements of the Weyl groups. They are found to give rise to the dynamics of various discrete integrable equations.
[1] Y. Shi, Translations in affine Weyl groups and their applications in discrete integrable systems, 63 pages, arXiv:2210.13736.
[2] H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Commun. Math. Phys. 220(2001), 165-229.

This talk focuses on q-Freud polynomials, orthogonal polynomials whose weight satisfies a q-difference Pearson relation. We investigate the behaviour of such polynomials in the cases q<1 and q>1. Using the RHP framework we deduce their asymptotic behaviour as their degree tends to infinity.

We consider a perturbed version of the second Painlevé equation (PII), which arises in applications, and show that it possesses solutions analogous to the celebrated Hastings-McLeod and tritronquée solutions of PII. The Hastings-McLeod-type solution of the perturbed equation is holomorphic, real-valued, and positive on the whole real line, while the tritronquée-type solution is holomorphic in a large sector of the complex plane. These properties also characterize the corresponding solutions of PII and are surprising because the perturbed equation does not possess additional distinctive properties that characterize PII, particularly the Painlevé property.

A brief review will be given of Lagrangian multiform theory, as the appropriate variational framework for systems integrable in the sense of multidimensional consistency. As a new result a Lagrangian 3-form structure will be presented for the generalised Darboux system, which encodes the Kadomtsev-Petviashvili (KP) hierarchy.

It was discovered how polynomial algebras appear naturally as symmetry algebra of quantum superintegrable quantum systems. They provide insight into their degenerate spectrum, in particular for models involving Painlevé transcendents for which usual approaches of solving ODEs and PDEs cannot be applied. Those algebraic structures extend the scope of usual symmetries in context of quantum systems, but they also been connected to different areas of mathematics such as orthogonal polynomials. Among them, the well-known Racah algebra which also admit various generalisations.
I will take a different perspective on those algebraic structures which is based on Lie algebras, their related enveloping algebras, partial Casimir and commutant. I will discuss how such approach differs from using differential operator realizations and why this framework offers advantages for their classification. I will point out as well how different methods from the study of Casimir invariant of non semisimple algebras which involves solving systems of PDEs can be applied in this context and greatly facilitate making calculations.
The talk will present various explicit examples, and in particular the symmetry algebra of the generic superintegrable systems on the 2-sphere which can be understood in a purely algebraic manner using an underlying Lie algebra.
The talk is based on following works:
R Campoamor-Stursberg, I Marquette, Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems, Annals of Physics 424 168378 (2021), arXiv 2020.168378
F Correa, MA del Olmo, I Marquette, J Negro, Polynomial algebras from su(3) and a quadratically superintegrable model on the two sphere, J.PhysA. Math. and Theor 54 015205 (2021) arXiv:2007.11163
R Campoamor-Stursberg, Ian Marquette, Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrodinger algebras, Annals of Physics. 437 168694 (2022) arXiv 2021.168694
D Latini, I Marquette, YZ Zhang, Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras, J.PhysA Math. and Theor. (2022) arXiv:2204.06840
Rutwig Campoamor-Stursberg, Danilo Latini, Ian Marquette, Yao-Zhong Zhang, Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras, arXiv:2211.04664

In this talk we show how to construct large classes of Lotka-Volterra ODEs in Rⁿ with n-1 first integrals. The building blocks we use will be Darboux Polynomials of the ODE. In the talk these concepts will be defined, and the procedure explained.

The Baxter-Fendley Z(N) model is a relatively simple N-state generalization of the quantum Ising chain. The energy eigenspectrum of this non-Hermitian model, subject to open boundary conditions, is composed of free parafermions, which are a natural generalization of the Ising free fermions to the complex plane. The model has remarkable physical properties, including boundary-dependent bulk behaviour, which is an example of the non-Hermitian skin effect. In this talk, based on new work with Alex Henry, I will discuss the appearance of exceptional points in the eigenspectrum.

While studying an integrable system associated with the Euler equation, Mishchenko and Fomenko (1978) produced a family of Poisson commutative subalgebras of the algebra of polynomial functions on a simple Lie algebra g. The quantization of such a subalgebra is a commutative subalgebra of the universal enveloping algebra U(g). We will discuss a method to construct these subalgebras of U(g) which is based on the corresponding affine vertex algebra at the critical level. The results are applied to calculate the Harish-Chandra images of the symmetrized basic invariants generating the center of U(g).

We study the geometry of confocal families on hyperboloids in pseudo-Euclidean spaces of dimension four in all signatures. The aim is to completely classify and describe them, and to prove Chasles' theorem in this ambient. The methodology we use includes concepts of pseudo-Euclidean and Euclidean geometry, and linear algebra as well. We also give a clue about the natural characteristics confocal families possess to be applied in a billiard theory. This research is done as a part of a Ph.D. project at the University of Sydney.

Noether's celebrated theorem reveals the deep connection between symmetries and integrals of a Hamiltonian system - they generate one another.  However, if the system is volume preserving and not Hamiltonian, no analogous result to Noether's Theorem has been shown. On 3-manifolds, we prove a Noether-type theorem - a volume preserving vector field will admit an integral if and only if it has a so-called conformal symmetry. To get the result, we require the existence of a one-form with a prescribed set of properties. We prove this one-form always exists in a region of positive measure called the toroidal region. The work is motivated by the study of magnetic fields designed for plasma confinement.

We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable factorization properties for row to row transfer matrices, allowing one to uniformly derive all functional relations for their eigenvalues and present the coordinate Bethe ansatz for the eigenvectors for all higher spin generalizations of the six-vertex model. The possibility of the Ising-type formulation of these models raises questions about the precedence of the traditional quantum group description of the vertex models. Indeed, the role of a primary integrability condition is now played by the star-triangle relation, which is not entirely natural in the standard quantum group setting, but implies the vertex-type Yang-Baxter equation and commutativity of transfer matrices as simple corollaries. As a mathematical identity the emerging star-triangle relation is equivalent to the Pfaff-Saalschuetz-Jackson summation formula, originally discovered by J. F. Pfaff in 1797. Plausibly, all vertex models associated with quantized affine Lie algebras and superalgebras can be reformulated as Ising-type models. (Based on the joint work with Sergey Sergeev, arXiv:2205.10708).

We study the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum. It turns out that they may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential equation of third order, for the surface representing the stratum. Integrable cases are isolated by requiring that the Gauss equation be compatible with another third-order hyperbolic differential equation. In particular, a variant of the integrable Tzitzeica equation is derived which encodes orthogonal coordinate systems on pseudospherical surfaces. This third-order equation is related to the Tzitzeica equation by an analogue of the Miura transformation.

We consider the nonlinear Schrödinger equation on the half-line x≥0 with a Robin boundary condition at x=0 and with initial data in the weighted Sobolev space H^1,1(R₊). We prove that there exists a global weak solution of this initial-boundary value problem and provide a representation for the solution in terms of the solution of a Riemann-Hilbert problem. Using this representation, we obtain asymptotic formulas for the long-time behavior of the solution, and results on the asymptotic stability of the stationary one-solitons under any small perturbation in H^1,1(R₊). In the focusing case, such a result was already established by Deift and Park using Bäcklund transformation, and our work provides an alternative approach to obtain such results. We treat both the focusing and the defocusing versions of the equation.