Integrable Systems 2020

The space of all harmonic maps from a 2-torus to the 3-sphere will be investigated, using spectral curves. I will explain Whitham deformations in this context and how they enable a proof that in an open and dense subset of a parameter space, the moduli space is smooth and has dimension two. The equivariant tori (spectral genus 0 or 1) can be described explicitly. The spectral curves of homogeneous tori form a disc. The genus one spectral curves densely foliate the parameter space with connected components which are either annuli or “helicoids”.

The Pearcey kernel is a classical and universal kernel arising from random matrix theory, which describes the local statistics of eigenvalues when the limiting mean eigenvalue density exhibits a cusp-like singularity. It appears in a variety of statistical physics models beyond matrix models as well.
In this talk, we consider the Fredholm determinant $\det\left(I-\gamma K^{\mathrm{Pe}}_{s,\rho}\right)$, where $0 \leq \gamma \leq 1$ and $K^{\mathrm{Pe}}_{s,\rho}$ stands for the trace class operator acting on $L^2\left(-s, s\right)$ with the classical Pearcey kernel. Based on a steepest descent analysis for a $3\times 3$ matrix-valued Riemann-Hilbert problem, we obtain asymptotics of the Fredholm determinant as $s\to +\infty$, which is also interpreted as large gap asymptotics in the context of random matrix theory.
This is a joint work with Shuai-Xia Xu and Lun Zhang.

For a Hamiltonian system with a periodic flow a natural object to consider is the space of its orbits. To construct integrable systems on these orbit spaces we study sets of functions with mutually vanishing Poisson brackets induced by separation of variables in the original space and pass to the quotient. Many integrable systems can be obtained in this way from super-integrable systems, for example the Manakov top. I will describe this construction starting with some simple super-integrable systems, and discuss some of the properties of the resulting integrable systems on the corresponding orbit spaces.

Wronskian polynomials of Hermite and Laguerre polynomials have interesting properties including curious root patterns. I will explain some of these properties and the consequences arising from their appearance in the sine-Gordon model at its free-fermion point.

The asymmetric simple exclusion process (ASEP) can be realised on the ring of multi-linear polynomials. We show how multi-species processes can be realised on higher degree polynomial spaces and how this connects to the theory of Macdonald and Koornwinder polynomials. As byproducts we obtain explicit matrix product formulas for Macdonald polynomials and a general method to construct duality functionals for stochastic exclusion processes from degenerations of multi-variable polynomials.

Topological recursion is a certain algorithm which computes a family of meromorphic multi-differentials on a given spectral curve. These multi-differentials are expected to contain the information of various enumerative invariants in mathematical physics. Many researchers, including Gukov, Sulkowski, Dumitrescu, Mulase, Bouchard, Eynard, discovered a relationship between the topological recursion and WKB analysis for Schrodinger-type differential equations. In this talk, I’ll explain these developments and an application to the WKB analysis of Painlevé equations.

It is a well-known fact that the integrable quad equations in the ABS list can be written as the derivative of a sum of three Lagrangian functions. In my talk I take the view that these Lagrangian functions are fundamental objects for discrete integrable systems, which may be used to construct (at least) two other types of integrable equations besides ABS equations. First, taking a sum of four Lagrangian functions that are defined on a vertex and its four nearest neighbours in the square lattice leads to five-point lattice equations, which for example include the equations known as discrete Laplace-type. The multidimensional consistency of these equations is proposed as consistency-around-a-face-centered-cube (CAFCC), which is an appropriate analogue of consistency-around-a-cube that is applicable to these equations. Second, taking a sum of four Lagrangian functions arranged in a square leads to quadrirational Yang-Baxter maps that have both two-component variables and parameters. Both of the above constructions lead to new equations that to the best of my knowledge were not previously considered in the context of discrete integrability. This talk is based on my recent works:
[1] “Two-component Yang-Baxter maps associated to integrable quad equations”, https://arxiv.org/abs/1910.03562
[2] “Interaction-round-a-face and consistency-around-a-face-centered-cube”, https://arxiv.org/abs/2003.08883

Exceptional orthogonal polynomials (EOP), classical orthogonal polynomials with gaps in their degree, have recently been the subject of flourishing new research connected to classical, univariate orthogonal polynomials. One area in particular is the connection between rational solutions to Painlevé equations and EOPs. In this presentation, we will discuss the connection between exceptional Hermite polynomials and rational solutions to the fourth Painlevé equation, which was previously known, and then extend the method to the sixth Painlevé equation via exceptional Jacobi polynomials. Throughout, we will see how superintegrable Hamiltonian systems and polynomial algebras play a role.

In this talk we will discuss how Darboux Polynomials (DPs) can be used to find/build rational integrals of ODEs. We will mainly discuss linear DPs.
First Reinout will give a brief introduction to Darboux Polynomials, and then discuss their application to some low dimensional integrable systems of Lotka-Volterra equations.
Then Peter will take over and extend the discussion to Lotka-Volterra equations in dimensions that can be arbitrarily large. The emphasis will be on establishing the superintegrability/Liouville integrability of these equations.

We study a family of planar area-preserving maps, which are linear on each of the right and left half-planes. Such maps can support quasiperiodic dynamics with a foliation of the plane by invariant curves. The parameter space is two dimensional and the set of parameters for which every orbit recurs to the half-plane boundary consist of algebraic curves, determined by the symbolic dynamics of the itinerary that connects boundary points. We study algebraic and geometric properties of these curves, in relation to such a symbolic dynamics.
This is joint work with Asaki Saito (Future University, Hakodate) and Franco Vivaldi (Queen Mary). A preprint with the same title will be posted to arXiv soon.

Eigenfunctions are shown to constitute privileged coordinates of
self-dual Einstein spaces with the underlying governing equation being revealed as the general heavenly equation. The formalism developed here may be used to link algorithmically a variety of known heavenly equations. We isolate a large class of self-dual Einstein spaces governed by a compatible system of dispersionless Hirota equations which is genuinely four-dimensional in that the metrics do not admit any conformal Killing vectors. In addition, Legendre transformations and connections with travelling wave reductions of the recently introduced TED equation which constitutes a 4+4-dimensional integrable generalisation of the general heavenly equation are discussed.

This discussion aims for postgraduate students, postdocs, and early-career researchers to discuss research and job opportunities for ECRs and aspects of the job application process. This discussion features a panel lead by Robert Marangell and Zsuzsanna Dancso (both University of Sydney).

This discussion aims to encourage students and ECRs to think about how to apply for research funding, even if it is a long way in the future. We ask participants to prepare a ~100-word research summary (to be used in a theoretical future grant application) for use in small groups. This discussion features a panel of Samuel Muller (Macquarie University) and Peter Forrester (University of Melbourne).