The asymmetric simple exclusion process (ASEP) is a model of randomly hopping particles in one spatial dimension. As well as being integrable, the ASEP enjoys very interesting scaling behaviour, with direct links to the Tracy--Widom distribution from random matrix theory. In this talk we will examine the problem of computing the transition probability between two different states of the system, in the case when all particles are distinguishable (i.e. they have different colours). The main tool for attacking this problem is a family of integrable stochastic vertex models, which serve as a discrete-time generalization of the ASEP.
This is based on joint works with Alexei Borodin, Jan de Gier and William Mead.

Conformal blocks are fundamental building blocks of conformal field theories and appear in the theory of Painleve equations through tau-functions, i.e the solutions of Painleve equations can be expressed in terms of conformal blocks. Such a connection is established through Fredholm determinant techniques.

In this talk, I will discuss recent work with Ian Marquette and Lisa Ritter on superintegable extensions of a Smorodinsky Winternitz potential associated with exception orthogonal polynomials (EOPs). EOPs are families of orthogonal polynomials that generalize the classical ones but with gaps in their degree sequence. The Hamiltonian systems associated with them will have integrals or motion that are higher-order in the momentum (or as differential operators in the quantum case). This allows for an interesting connection with Painleve equations, both recovering known rational solutions to Painleve equations and pointing to new forms of rational solutions to P_VI via Jacobi polynomials.

Although the notion of singularity confinement was first introduced as a crucial attribute of the singularities of the discrete KdV (dKdV) equation, as of yet there is still no rigorous definition of the notion of `confinement' in the context of lattice equations. In fact, somewhat ironically, it has taken nearly 30 years before an exhaustive study of the singularities of the dKdV equation finally revealed their intriguing properties and the full richness of the singularity structures they produce.
In this talk I will explain the basic classification of singularities that arise in case of the dKdV equation, in four distinct classes, one of which consists of a novel `strip-like' type of singularity. These strips or bands can interact with the singularities in the other three classes, thereby producing singularity patterns of increasing complexity. Moreover, I will show that the interaction of these singularities with a particular type of singularity can be described by means of a special symbolic dynamics which turns out to be intimately related to Box&Ball dynamics.
If time permits I will show examples of similar dynamics that arise in the case of a modified discrete KdV equation.
This talk is based on the paper "On the singularities of the discrete Korteweg–de Vries equation", D. Um, A. Ramani, B. Grammaticos, R. Willox & J. Satsuma, J. Phys. A: Math. Theor. 54 (2021) 095201.

The Lagrange Top (heavy symmetric rigid body with a fixed point) with an additional quadratic potential is described in global coordinates using a 7-dimensional Poisson structure. The set of critical values of the energy-momentum map has a rational parametrisation that is derived from the global description. The image is not simply connected, describes super- and sub-critical Hopf bifurcations, and has Hamiltonian monodromy in the action variables. Our main observation is that upon quantisation this system is equivlalent to the most general confluent Heun equation. The Heun equation is the most general Fuchsian equation with 4 regular singular points, generalising the hypergeometric equation, and a degeneration of this equation (with non-regular singular points) is the confluent Heun equation, also known as the generalised spheroidal wave equation. We conclude that the generalised spheroidal wave equation exhibits quantum monodromy in its joint spectrum.
Joint work with Sean R. Dawson and Diana M.H. Nguyen, University of Sydney

Two classical but perhaps little known facts of "elementary" geometry are that an ellipsoid may be sliced into two one-parameter families of circles and that ellipsoids may be deformed into each other in such a manner that these circles are preserved. In fact, as an illustration of these remarkable properties of ellipsoids, Hilbert and Cohn-Vossen present pictures of deformable wooden models of sliced ellipsoids in their classical monograph Geometry and the Imagination. The purpose of this talk is to demonstrate that recently introduced concepts in (integrable) Discrete Differential Geometry allow one to discretise ellipsoids in such a manner that a variety of classical properties such as those mentioned above survive. Connections with (discrete) confocal coordinate systems are explored.

Formulation of the Painleve equations and their generalisations as birational representations of affine Weyl groups provides us with an elegant and efficient way to study these highly transcendental, nonlinear equations. In particular, it is well-known that discrete evolutions of Painleve equations are given by translational elements of extended affine Weyl groups.
On the other hand, theory of the Weyl group, or in general Coxeter group is a classical area of algebra rich with many remarkable properties. Characterisation of integrable systems in terms of such groups naturally leads us to the question: how much of the intrinsic properties of the group remain in the realisations of integrable equations and what are their implications in this particular context?
It has been found that certain non-translational elements of infinite order of an affine Weyl group can also give rise to discrete Painleve equations. We clarify the nature of these elements in terms of Brink-Howlett theory of normalizers of Coxeter groups. More generally, we show that varieties of integrable systems relate in a way that is consistent with the different subgroup structures arising from the theory of normalizers of Coxeter groups.

At the heart of an integrable discrete map is the existence of a sufficient number of integrals of motion. When the map is birational and the integral is assumed to be a rational function of the variables, many results from algebraic geometry and number theory can be employed in the quest to find integrals. I will review some recent contributions in this area.

This talk summarizes joint work with D.J. Zhang, D.D. Zhang and X. Wei, on multi-component extensions of CAC systems, how to obtain auto-Backlund transformations from auto-Backlund transformations, and torqued ABS equations, see papers 33, 37, and 40 from https://wiskun.de/publications.

In this talk I will report a detailed study of the scaling limit of a certain critical, integrable inhomogeneous six-vertex model subject to twisted boundary conditions. It is based on a numerical analysis of the Bethe ansatz equations as well as the powerful analytic technique of the ODE/IQFT correspondence. The results indicate that the critical behaviour of the lattice system is described by the gauged SL(2) WZW model with certain boundary and reality conditions imposed on the fields. Our proposal revises and extends the conjectured relation between the lattice system and the Euclidean black hole non-linear sigma model that was made Ikhlef, Jacobsen and Saleur in the 2011.