Ninth Workshop on Integrable Systems
The University of Sydney
2–3 December 2021
8:50  9:00  Nalini Joshi Welcome 
9:00  9:40 
Michael Wheeler
Transition probabilities in the asymmetric simple exclusion process
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 TracyWidom 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 discretetime generalization of the ASEP.

9:50  10:20  coffee break & discussion 
10:20  11:00 
Harini Desiraju
Conformal blocks on a torus via Fredholm determinants
Conformal blocks are fundamental building blocks of conformal field theories and appear in the theory of Painleve equations through taufunctions, i.e the solutions of Painleve equations can be expressed in terms of conformal blocks. Such a connection is established through Fredholm determinant techniques. 
11:10  11:50 
Sarah Post
Rational extensions of superintegrable systems, exceptional polynomials and Painleve equations
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 higherorder 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. 
12:00  13:00  lunch break 
13:00  13:40 
Ralph Willox
The singularity structure of integrable lattice equations
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.

13:50  14:30 
Holger Dullin
The Lagrange top and the confluent Heun equation
The Lagrange Top (heavy symmetric rigid body with a fixed point) with an additional quadratic potential is described in global coordinates using a 7dimensional Poisson structure. The set of critical values of the energymomentum map has a rational parametrisation that is derived from the global description. The image is not simply connected, describes super and subcritical 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 nonregular 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.

14:40  15:10  afternoon tea & discussion 
9:00  9:40 
Wolfgang Schief
A canonical discrete analogue of the classical circular cross sections of ellipsoids and their isometric deformation
Two classical but perhaps little known facts of "elementary" geometry are that an ellipsoid may be sliced into two oneparameter 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 CohnVossen 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. 
9:50  10:30 
Yang Shi
Normalizer theory of Coxeter groups and discrete integrable systems
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 wellknown that discrete evolutions of Painleve equations are given by translational elements of extended affine Weyl groups.

10:40  11:20 
John Roberts
On finding integrals in birational maps
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. 
11:30  12:30 
Poster session
Sean Dawson, Diana Nguyen, Holger Dullin Monodromy in prolate spheroidal harmonics (PDF, 2.9M) Hammed Fatoyinbo Solitons: ballbox model approach (PDF, 1.3M) Giorgio Gubbiotti Coalgebra symmetry for discrete systems (PDF, 0.1M) Josh Holroyd Asymptotic analysis and special solutions of a family of Painlevelike equations (PDF, 0.3M) Hirotaka Iino, Ralph Willox Discretization of integrable subcases of the HenonHeiles system and the Lorenz system (PDF, 0.2M) Sakkaravarthi Karuppaiya Dynamics of solitons, breathers, and rogue waves in inhomogeneous media: similarity transformation to explore variablecoefficient nonlinear model equations (PDF, 10M) Tomas Lasic Latimer On the qRiemannHilbert problem and its application to qorthogonal polynomials (PDF, 0.1M) Fahimeh Mokhtari Normal form for completely integrable triplezero singularities (PDF, 0.1M) Diana Nguyen, Sean Dawson, Holger Dullin Integrable systems arising from separation on S^{3} (PDF, 1.3M) Pieter Roffelsen Monodromy surfaces of qPainleve equations (PDF, 0.1M) Andreas Vollmer Superintegrable Hamiltonian systems on conformal manifolds (PDF, 0.2M) 
12:30  13:30  lunch break 
13:30  14:10 
Peter van der Kamp
On CAC and Backlund transformations
This talk summarizes joint work with D.J. Zhang, D.D. Zhang and X. Wei, on multicomponent extensions of CAC systems, how to obtain autoBacklund transformations from autoBacklund transformations, and torqued ABS equations, see papers 33, 37, and 40 from https://wiskun.de/publications. 
14:20  15:00 
Vladimir Bazhanov
Scaling limit of the sixvertex model and twodimensional black holes
In this talk I will report a detailed study of the scaling limit of a certain critical, integrable inhomogeneous sixvertex 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 nonlinear sigma model that was made Ikhlef, Jacobsen and Saleur in the 2011. 
15:10  15:40  closing & discussion 