Participants

General relativity is Albert Einstein’s beautiful theory which combines the space-time continuum, its geometry and its interaction with matter into a unified notion of gravity. It is currently the best experimentally confirmed theory which describes our cosmos as a whole, as well as astrophysical systems consisting of black holes and/or neutron stars. However, even 100 years after its invention, there are still fundamental outstanding theoretical questions: For example, regarding the existence and properties of “singularities” within this theory, whether these “singularities” are always hidden inside “horizons”, or, whether “time travel” is, in principle, possible. In this talk, I will discuss some of these issues in the context of cosmology. Many of these problems turn out to be tremendously complicated mathematically — and we therefore often do not even know how to attack the beast. Having said this, it is an exciting prospect that the recent experimental discovery of gravitational waves may allow us to test some of the resolutions to such problems in the not so distant future.

Wronskian determinants of classical orthogonal polynomials have a number of applications in mathematical physics. For example, they are

• building blocks of certain rational solutions of the Painleve equations;
• the main classes of examples of exceptional orthogonal polynomials;
• certain solutions of the Kadomtzev-Petviashvili hierarchy of partial differential equations in integrable systems.

I will describe how Wronskian Hermite and Wronskian Laguerre determinants describe the excited-state ordinary differential equation/integrable model correspondence at the free-fermion point of the sine-Gordon model. We classify the set of allowed Wronskians and describe some interesting properties of the distribution of their zeros, the latter being one of the open mathematical questions concerning Wronskian determinants of classical orthogonal polynomials.

I will discuss some recent work with Fei Han on T-duality in an H-flux, that builds on earlier work with Bouwknegt and Evslin which will also be reviewed. The novelty in the approach is that it uses loop space in an essential way.

A systematic method to construct Bethe type eigenstates of quantum integrable models without U(1) symmetry will be presented. Usually such kind of models can not be solved via conventional algebraic Bethe Ansatz due to absence of an obvious reference state. In this talk, I shall show that with the eigenvalues obtained via the off-diagonal Bethe Ansatz and a proper representation basis, the corresponding eigenstate for a given eigenvalue can be retrieved. A general method for constructing representation basis of high-rank integrable models will also be introduced.

In recent years, many nonperturbative methods have been developed which allow the calculation of dynamical properties of integrable systems of relevance for experiments among others on magnetic systems and cold atoms. These systems exhibit relaxation and equilibration behaviour which cannot be simply described by traditional textbook methods, and can lead to long-lived non-thermal equilibrium states. This talk will provide an overview of recent research in this area, and introduce recently-discovered methods to treat quenched and driven systems in various experimentally-relevant contexts.

For over forty years Supergravity, the quantum field theory joining general relativity with local supersymmetry, has been at the heart of the still elusive search for a unified framework for gauge and gravitational interactions. It has brought us to explore and often challenge the most profound ideas in theoretical physics, both in relation with string theory and on its own ground. Moreover, it has naturally driven us to delve into beautiful concepts in pure mathematics, also generating new directions. We shall present a personal panorama that emphasises the geometric approach to supergravity, visit interesting mathematical tools encountered on the way and reflect on where we stand.

We first study the p+ip Hamiltonian isolated from its environment (closed model) through the Bethe Ansatz solution and consider the case of a large particle number. A continuum limit approximation is applied to compute the ground-state energy. We discuss the evolution of the solution curve, and the limitations of this approach. We then consider an alternative approach that transforms the Bethe Ansatz equations to an equivalent form in terms of the real-valued conserved operator eigenvalues. This approach also generalizes to accommodate interaction with the environment (open model)

Exclusion process has been the default model for transportation phenomenon. One fundamental issue is to solve the master equation analytically, which, in principle, gives all the correlation function of the system. In this talk, we will focus on an exclusion process with 2 species particles: the AHR (Arndt-Heinzl-Rittenberg) model. We will give a full derivation of its Green’s function as well as its joint current distributions. We will also study its long time behaviour with step type initial conditions.

We study two specific lattice walk model, Reverse Kreweras Walk and Kreweras Walk in this work. The walk is restrict to the first quadrant and we assign different weights on steps visiting x axis, y axis and origin. The interaction are still homogeneous on each boundary. These two cases are only partially solved in a recent work by Beaton, Owczarek and Rechnitzer (2018). We will walk through the algebraic kernel method and completely solve these two models here. Without weights, these two walks have algebraic generating functions. With the interaction, the generating function of Reverse Kreweras Walk is still algebraic but the generating function of Kreweras Walk is D-finite.

A model of dipolar bosons confined to three coupled wells will be introduced and analysed. The main result of this research is a protocol for achieving controlled tunneling, via analytic formulae for frequency and amplitude, through variation of an external field. The role and consequences of integrability of the system will be discussed.

We compute the free energy and surface tension function for the five-vertex model, a model of non-intersecting monotone lattice paths on the grid in which each corner gets a positive weight. We give a variational principle for limit shapes in this setting, and show that the resulting Euler-Lagrange equation can be integrated, giving explicit limit shapes parameterized by analytic functions.

We study the motion of bound null geodesics with fixed coordinate radius around a five-dimensional rotating black hole. These spherical photon orbits are not confined to a plane, and can exhibit interesting quasiperiodic behaviour. In this talk we will discuss necessary conditions for such orbits, and plot representative examples of some of the types of possible orbits, commenting on their qualitative features.

In recent years it has become apparent that there is an exactly solved Z(N) model described by free parafermions. For N = 2 this model reduces to the Ising model. I will discuss ongoing work on this model, particularly with regard to what can be said about spin correlations. This talk is based on recent work with Zi-Zhong Liu and Huanqiang Zhou in Chongqing and Alex Henry at ANU.

Distinguished curves play an important role in the study of many geometries and are of interest for a wide range of applications. Projective and conformal geometry are two examples of parabolic geometries and thus possess natural notions of distinguished curves. We will present a characterization of distinguished curves in these settings phrased in the language of tractor calculus, a framework well suited to the study of parabolic geometries. Conserved quantities along curves also fit nicely into the tractor description giving possible applications to integrable systems and the study of PDEs. This is joint work with A. Rod Gover (University of Auckland) and Arman Taghavi-Chabert (American University of Beirut).

This talk is a pedagogical review of recent works arXiv:1809.00802, arXiv:1807.09098 and arXiv:1805.08055.

We revise the solutions of the forced Korteweg – de Vries equation describing the resonant interaction of a solitary wave with external pulse-type perturbations. In contrast to the previous works, where only the limiting cases of very narrow forcing of delta-function type or very narrow solitary wave in comparison with the width of external perturbation were studied, we consider here an arbitrary relationship between the widths of a soliton and external perturbation. We show that in the number of particular cases the exact solutions of the forced Korteweg – de Vries equation can be obtained. We use the earlier derived approximate set of equations up to the second-order accuracy on small parameter epsilon characterising the amplitude of external force and analyse its exact (where possible) and numerical solutions. Theoretical results obtained by asymptotic method are compared with the results of direct numerical modelling within the forced Korteweg-de Vries equation.

Applying a Newtonian form of Schrodinger’s equation to the potential wells of large, gravitationally-bound structures such as galaxies leads to a complete set of weak-field gravitational eigenstates. These can be used to describe the interactive behaviour of elementary uncoalesced particles using their quantum wavefunctions rather than the traditional approach of classical mechanics (Ernest,2009 a, b). One expects quantum and classical approaches to predict the same interaction behaviour, but in large-scale, galactic and super-galactic gravitational wells, quantum theory predicts the existence of an overwhelming number of well-bound, long-lived and weakly interacting ‘dark’ gravitational eigenstates, analogous to the Rydberg states of atoms. Ordinary particles such as electrons, that normally make themselves visible via optical scattering processes, will have reduced scattering cross sections if their wavefunctions contain significant numbers of these dark states in their eigenspectra. The traditionally accepted cross sections used for density calculations would consequently be wrong on galactic and super-galactic scales resulting in the true number densities of particles being much greater than what we observe them to be. Ordinary particles can thus function as the much sort-after ‘dark matter particle’, and hence provide a resolution to the dark matter problem without a need to invoke the ad hoc existence of specialized exotic, and as yet unverified, particles. Dark matter is then seen to be simply ordinary elementary particles having “dark eigenspectral disguises”, while all the time preserving the evidence for it obtained from the cosmic microwave background and big bang nucleosynthesis. This talk will describe the quantum theory of dark matter, outline its successes, and discuss some of the mathematical difficulties in making accurate numerical predictions that are observationally testable. A. D. Ernest, “Gravitational eigenstates in weak gravity: I. Dipole decay rates of charged particles”, 2009a, J. Phys. A: Math. Theor. 42, 115207 A. D. Ernest, “Gravitational eigenstates in weak gravity: II. Further approximate methods for decay rates”, 2009b, J. Phys. A: Math. Theor. 42, 115208

In three dimensions, the conformal structure of spacetime is encoded within the Cotton tensor. In particular, spacetime is conformally flat if and only if the Cotton tensor vanishes. The latter has a deep connection with the action of conformal gravity, which plays a major role in the construction of topologically massive gravity; a theory that describes the massive spin-2 graviton. By employing a modern approach to conformal gravity as a gauge theory, we construct higher-spin analogues of the linearised Cotton tensor in any conformally flat space. Using these higher-spin Cotton tensors, we formulate higher-spin extensions of topologically massive gravity in both Minkowski and anti-de Sitter space.

It has been known for some time now (Ernest, 2009a; Ernest, 2009b) that the complete set of gravitational eigenstates, obtained by applying quantum mechanics to galactic and super-galactic gravitational wells, includes an overwhelming number of well-bound, long-lived and weakly interacting ‘dark’ gravitational eigenstates, analogous to the Rydberg states of atoms. Ordinary particles such as electrons, that normally make themselves visible via optical scattering processes, would have reduced scattering cross sections if their wavefunctions contained significant numbers of these dark states in their eigenspectra. This would result in the number densities of particles being much greater than what we observe them to be, because the traditionally accepted cross sections used for density calculations would be wrong on galactic and super-galactic scales. It means that dark matter could simply be ordinary elementary particles having a “dark eigenspectral disguise”, while all the time preserving the evidence for it obtained from the cosmic microwave background and big bang nucleosynthesis. In this talk we present a continuation of the study (Whinray and Ernest, 2018) of the variation in eigenstate-interaction properties with their quantum parameters, in large gravitational systems. If the trends in the properties of gravitational eigenstates continue to behave as expected then it may well be that the high quantum-numbered, Rydberg-type gravitational eigenstates could enable ordinary particles to function as the much sort after ‘dark matter particle’, and hence provide a resolution to the dark matter problem without a need to invoke the ad hoc existence of specialized exotic, and as yet unverified, particles.

The celebrated Haldane–Shastry model is a long-range spin chain enjoying quantum-affine symmetries already at finite system size. Its q-deformation, i.e. partially isotropic or XXZ-like version, was found by D. Uglov back in ’95 in an e-print that seems to have gone by unnoticed. Recently we managed to express this Hamiltonian in a more friendly, pairwise form that makes hermiticity for real q manifest. After reviewing the model I will present its — for now conjectural — exact (highest-weight) eigenvectors for finite size. With respect to a modification of the coordinate basis involving R-matrices, the wave functions are a Vandermonde factor squared times a Macdonald polynomial (with t=q^2). This is ongoing work together with V. Pasquier and D. Serban.

The G-function technique to compute the spectrum of unbounded operators in Bargmann space connects the singularity theory of ordinary differential equations in the complex domain with discrete symmetries of the Hamiltonian. Examples from quantum optics are presented where spectral determinants can be computed in terms of known transcendental functions. Applications to more general monodromy problems are briefly outlined.

In this talk I will discuss some ongoing work describing a possible link between non-isometric T-duality and Poisson–Lie T-duality.

The coupling time of the stochastic Ising model is the random time required for processes started in the all-plus and all-minus states to coalesce, when coupled in the natural way. We show that, for any dimension d, and sufficiently high temperature, the (appropriately standardised) coupling time of the stochastic Ising model on d-dimensional tori tends weakly to a Gumbel distribution.

The duality concept from string theory is very powerful: there are two pictures of a single phenomenon, each with hard and easy parts. Furthermore, the duality swaps hard with easy so that it suffices to look at the easy parts of each side to obtain the full picture. At the level of functions, the Fourier transform provides such a duality. I will explain how T-duality, which is a geometric Fourier transform, arises in solid state physics and how it led to the discovery of crystallographic T-duality by K. Gomi and myself. This gives new computational tools for twisted K-theory.

We are interested in some properties of the self-avoiding walk (SAW) process on the complete graph. In particular, we will be looking for how quickly the expectation and variance of walk length grows with the number of vertices, around a pseudocritical point. Yadin’s paper in 2014 finds these quantities about a critical point. In this article, we will observe the behaviour of these quantities about a critical window.

Fractional-level WZW models are expected to play an important role in understanding general classes of logarithmic CFTs. Whilst the rank-1 fractional-level models (ie those based on sl₂ and osp(1|2) are now fairly well understood, the situation for higher ranks is far from satisfactory. In this talk, I will discuss recent results for the model based on sl₃ at level -3/2 including a description of the spectrum, characters, modular transformations and (if time permits) fusion rules.

In this talk we consider solutions to the reflection equation related to the higher spin 6-vertex model. We derive difference equations for the matrix elements of the K-matrices for arbitrary spin s and solve them in terms of hypergeometric functions.

The self-avoiding walk (SAW) on Z^d is known to display the same large-scale behaviour as simple random walk (SRW) when d is large. In particular, the SAW and SRW Green’s functions are known to display the same asymptotics when deg = 5. On finite boxes, however, where SAWs must have finite length, this SAW-SRW correspondence breaks down. To recover such a correspondence, we introduce a random-length random walk (RLRW), and rigorously derive its Green’s function. By combining these general RLRW results with the asymptotic walk-length distribution of the SAW on the complete graph, we obtain an explicit conjecture for the SAW Green’s function on high-dimensional tori, both at criticality and within a broad family of scaling windows. Using a random walk representation of the Ising model, due to Aizenman, these conjectures extend naturally to the Ising model, where they then shed light on a number of questions under current debate by computational/theoretical physicists. Finally, we will also discuss the case of free boundary conditions, where the RLRW model again clarifies a number of actively debated questions.

Advances in the application of atomic force microscopy to manipulate single polymers motivates new work on lattice polymers subject to a force. We study uniform 3-star polymers with one branch tethered to an attractive surface and another branch pulled by a force away from the surface. Each branch of the 3-star lattice is modelled as a self-avoiding walk on the simple cubic lattice with one endpoint of each branch joined at a common node. Recent theoretical work by Janse van Rensburg and Whittington (JPA 2018) determined the phase diagram but can only be specific in asymptotic regimes. We investigate this system by using the flatPERM Monte Carlo algorithm with special restrictions on the endpoint moves to simulate 3-stars up to branch length 128. We provide numerical evidence of the four phases and in particular that the ballistic-mixed and adsorbed-mixed phase boundaries are first-order transitions. The position of the ballistic-mixed and adsorbed-mixed boundaries are found at the expected location in the asymptotic regime of large force and large surface-monomer interaction energy. These results indicate that the flatPERM algorithm is suitable for simulating star lattice polymers and opens up new avenues for numerical study of non-linear lattice polymers. In particular an interesting question is how do pulled f-stars behave in two-dimensions, for example on the square lattice, where the adsorption of one branch is screened by another, preventing adsorption of the entire f-star to the surface. Numerical work can provide new insights into these systems.

In this talk, we discuss the study of the N=2 and affine osp(1|2) minimal models at admissible levels using the method of coset construction. These sophisticated minimal models are rich in mathematical structure and come with various interesting features for us to investigate. The N=2 minimal model, tensored with a free boson, can be extended into a sl₂ minimal model tensored with a pair of fermionic ghosts, whereas a osp(1|2) minimal model is an extension of the tensor product of certain Virasoro and sl₂ minimal models. We can therefore induce the known structures of the representations of the coset components and get a rather complete picture for the minimal models we want to investigate.The main outcome of the project includes a classification of the irreducible modules, their characters and fusion rules.

We investigate various aspects of the structure and representations of the Z₂×Z₂-graded Lie colour algebra, denoted gl(m₁,m₂|n₁,n₂).

The paradigm of investigating non-equilibrium phenomena by considering stationary states of emergent hydrodynamics has attracted a lot of attention in the last years. Recent proposals of an exact approach in integrable cases, making use of TBA techniques, are presented and discussed.