Eleventh Workshop on Integrable Systems
The University of Sydney
30 November – 1 December 2023
9:30  9:40  Nalini Joshi Opening 
9:40  10:10 
Wolfgang Schief
The differential geometry of the multidimensionally consistent
TED equation
The notion of multidimensional consistency has proven to be central in both the algebraic and geometric theory of discrete integrable systems. In this talk, we present a natural differentialgeometric interpretation of a highly symmetric 4+4dimensional dispersionless integrable differential equation. The multidimensional 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. 
10:20  10:50 
Peter van der Kamp
Darboux polynomials, trees and LotkaVolterra systems
We explore how a tree on n vertices, through an associated weighted complete digraph, relates to a superintegrable ncomponent 3n2 parameter family of LotkaVolterra systems. These systems are measurepreserving, as is their
Kahan discretisation. 
11:00  11:30  coffee break & discussion 
11:30  12:00 
Reinout Quispel
Discretizing ODEs while preserving integrals/integrability
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. 
12:10  12:40 
Ian Marquette
Polynomial algebras from Lie algebra reduction chains g⊃g'
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 threegenerator
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
centrallyextended Schrödinger algebra in (3+1)dimensional spacetime.
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.

12:50  14:00  lunch break 
14:00  14:30 
Alessandro Sfondrini
Integrability on the string worldsheet
I will give a pedagogical review of how integrablemodels techniques can be used to perform exact computations in string theory, by treating the string worldsheet model as a twodimensional 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. 
14:40  15:10 
JeanEmile Bourgine
A (q,t)deformation of the Toda integrable hierarchy
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 differencedifferential 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 Rmatrix of the toroidal algebra. 
15:20  16:00  afternoon tea & discussion 
16:00  16:30 
Pieter Roffelsen
Singularities of Painlevé functions, Heun equations and generalised Hermite polynomials
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). 
16:40  17:10 
Harini Desiraju
Orthogonal polynomials on elliptic curves and Painlevé VI equation
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 RiemannHilbert 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). 
9:00  9:30 
Yousuke Ohyama
Nonlinear and linear connection problems on qPainlevé equations
The RiemannHilbert correspondence is a powerful tool to study the Painlevé differential equations. We study Birkhoff’s version on qanalogue of the RiemannHilbert problem and the RiemannHilbertBirkhoff correspondence is also important to study the qPainlevé equation. We mainly study the sixth qPainlevé equation and we may also study the fifth qPainlevé equation. 
9:40  10:10 
Andrew Kels
Consistency for 5point lattice equations
In this talk I will present formulations of consistency for 5point lattice equations, particularly in lattices of type D and honeycomb lattices. The consistency property may be used to derive Lax matrices for 5point equations. Known examples of consistent equations in both types of lattices give quadratic growths of degrees of iterates, indicating they are integrable. 
10:20  10:50 
John Roberts
Sfractions on hyperelliptic curves and integrable Volterra maps
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. 
11:00  11:30  coffee break & discussion 
11:30  12:00 
Dmitry Demskoy
The lattice SineGordon equation as a superposition formula for an NLStype system
Treating the lattice sineGordon equation, along with its two simplest generalized symmetries, as a compatible system allows one to investigate another integrable system not directly connected to the sineGordon equation: a modified nonlinear Schrödinger system with derivative. 
12:10  12:40 
Renjie Feng
Extreme gap problems for classical random matrices
I will first introduce two types of random matrices and discuss classical results such as the semicircle law and the TracyWidom law. Then I will present our recent findings regarding extreme gap problems in classical random matrices and propose several conjectures. 
12:50  14:00  lunch break 
14:00  14:30 
Emma Carberry
Constant mean curvature tori in R^{3} with lowest spectral genus
A constant mean curvature (cmc) torus in R^{3} 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 R^{3} as an isoceles rightangled triangle, with the Wente family along its diagonal. Furthermore, the boundary of this triangle contains solutions to a number of different integrable systems: sinhGordon, NLS, KdV and nonconformal harmonic maps all make an appearance. 
14:40  15:10 
Sean Gasiorek
Dynamics and periodicity conditions for the integrable Boltzmann system
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, Cayleytype periodicity conditions, and more. This is joint work with Milena Radnović (University of Sydney). 
15:20  15:50  closing, afternoon tea & discussion 