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Fahad Sameer Alshammari	Casimir Operators of conformal Galilie algebra
Michael Assis	Exactly Solving Regular Planar Vertex Models With Dimers
	New exact solutions of vertex models on the honeycomb, square, and triangular lattices will be presented, along with some of their applications. On the honeycomb lattice the result is related to the dilute O(1) loop model, the O(1) fully packed loop model, dimers packings, and the anisotropic Ising model on the triangular lattice. The new square lattice vertex model is related to origami folding lattices, such as the Miura-ori and Huffman lattice, and through a duality relation, to the known free-fermion 8-vertex model. New vertex models on the triangular lattice likewise have similar interpretations. A general technique for solving the (conjectured) most general dimer-solvable vertex models on regular planar lattices will be presented.
Murray Batchelor
Vladimir Bazhanov	Towards exact solution of non-linear sigma models
Nicholas Beaton
Andrea Bedini
Debra Bernhardt (Searles)	Equilibrium distributions as dissipationless states
Raphael Boll	On the variational formulation of discrete hyperbolic equations
	We present the variational structure of certain discrete hyperbolic equations by means of their pluri-Lagrangian formulation, namely, of the (two-dimensional) quad-equations from the well-known ABS list and of the (three-dimensional) discrete KP equation. In both cases, we consider the elementary building blocks of the Euler-Lagrange equations, the so-called corner equations, which are the Euler-Lagrange equations on the manifolds consisting of the facets of one cube or hypercube, respectively. However, the differences in the variational formulation of these two classes of equations are rather surprising: the corner equations on one cube in the twodimensional case are consistent, in the sense that they have the have the minimal possible rank 2. Moreover, their set of solutions is essentially bigger than the set of solutions of quad-equations. In contrast, the corner equations on one hypercube in the three-dimensional case are not consistent in the manner described above, but they are, in a sense, equivalent to the consistent system of dKP equations on the hypercube. Joint work with Matteo Petrera and Yuri B. Suris .
Jon Borwein
Gary Bosnjak
Mireille Bousquet-Mélou	The Potts model on planar maps (keynote)
	Let q be an integer. We address the enumeration of q-colored planar maps (planar graphs embedded in the sphere), counted by the total number of edges and the number of monochromatic edges (those that have the same colour at both ends). In physics terms, we are averaging the partition function of the Potts model over all maps of a given size. We prove that the associated generating function is algebraic when q is of the form 2 + 2 cos(j_/m), for integers j and m (but distinct from 0 and 4). This includes the two integer values q = 2 and q = 3, for which we give explicit algebraic equations. For a generic value of q, we prove that the generating function satisfies an explicit system of differential equations. Both results hold as well for planar triangulations, with a strikingly similar system of differential equations. The starting point of our approach is a recursive construction of q-coloured maps, in the spirit of what W. Tutte did in the seventies and eighties for properly coloured triangulations. This model has also been addressed by other authors and other methods (Bonnet & Eynard in 1999, and more recently Guionnet, Jones, Shlyakhtenko & Zinn-Justin, and Borot, Bouttier & Guitter), but our results are of a different nature and more explicit. Joint work with Olivier Bernardi, Brandeis University .
Peter Bouwknegt	Towards a classification of topological phases for strongly interacting systems
Richard Brak	Coxeter groups, Polymers and Random Walks in Restricted Geometries
Richard Brent
Elliot Catt
Yao-ban Chan	Upper bounds on the growth rate of hard squares and related models via corner transfer matrices
	In this talk, we study a problem motivated by data storage in a two-dimensional array. Hardware considerations induce constraints in the possible configurations of magnetic spins in this array, and the storage capacity can be found by modelling the constraints as lattice spin models. In a previous ANZAMP talk, we described how to calculate lower bounds on the capacities of several such constraints. Here, we use an assortment of techniques from combinatorics, statistical mechanics and linear algebra to derive the corresponding upper bounds. Our method starts from Calkin and Wilf's transfer matrix bound, then bounds that with the Collatz-Wielandt eigenvalue bound. To achieve a tight bound, we use Baxter's corner transfer matrix ansatz combined with Nishino and Okunishi's corner transfer matrix renormalisation group method. This results in an algorithm which does not require exponential memory and is easy to parallelise, allowing us to make dramatic improvements to the best known upper bounds.
Parisa Charkhgard
Zeying Chen	Duality for the asymmetric exclusion process
Nathan Clisby	Monte Carlo calculation of a new surface amplitude ratio for self-avoiding walks
	We describe Monte Carlo simulations of self-avoiding walks, half-space walks and bridges which have led to accurate estimates of a new universal amplitude ratio, and the growth constant for various lattices. This universal amplitude ratio is interesting because it involves walks attached to a surface. Related universal quantities can be derived for other geometries and simple random walks.
Jan De Gier	N=2 supersymmetry on the lattice without fermion conservation
	I will discuss an integrable supersymmetric fermion chain without fermion conservation. The spectrum of this model displays a very large degeneracy at all levels. I will discuss the symmetries of the model explaining these degeneracies, including a mechanism for the creation an annihilation of zero energy pair-excitations, much like Cooper pairs.
Silvestru Sever Dragomir	Sharp Bounds for Quantum f-Divergence
	The concepts of quantum f-Divergence and, in particular relative entropies of various sorts, play an important role in different subjects, such as statistical mechanics, information theory, dynamical systems, ergodic theory, biology, economics, human and social sciences. In Quantum Mechanics, they are closely related to the problem of the quantification of entanglement, the distinguishability of quantum states and to thermodynamical ideas. In this presentation we give some inequalities for quantum f-divergence of trace class operators in Hilbert spaces. It is shown that for normalised convex functions it is nonnegative. Some upper bounds for quantum f-divergence in terms of variational and \chi^2-distances are provided. Applications for some classes of divergence measures such as Umegaki and Tsallis relative entropies are also given.
Eren Metin Elci	Bridges in the random-cluster model
	The random-cluster model, a correlated bond percolation model, uni_es a range of important models of statistical mechanics in one description, including independent bond percolation, the Potts model and uniform spanning trees. By introducing a classi_cation of edges based on their relevance to the connectivity we study the stability of clusters in this model. We prove several exact relations between the different edge types for general graphs that allow us to derive unambiguously the _nite-size scaling behavior of the density of bridges and non-bridges. For percolation, we are also able to characterize the point for which clusters become maximally fragile and show that it is connected to the concept of the bridge load. Combining our exact treatment with further results from conformal _eld theory, we uncover a surprising behavior of the (normalized) variance of the number of (non-)bridges, showing that it diverges at criticality in two dimensions below or at the value 4cos^2(\pi/\sqrt{3}) = 0.2315891... of the cluster weight q.†
Rasul Esmaeilbeigi
Denis Evans	Dissipation and the foundations of classical statistical thermodynamics (keynote)
	In this talk I will discuss the derivation of the postulates of statistical mechanics from the laws of mechanics and the axiom of causality ñ that cause precedes effect. In order to do this we derive three main theorems: the Fluctuation Theorem [1] that gives the relative probability that path integrals of the dissipation take on opposite values, ± A; the Dissipation Theorem [2] that relates nonequilibrium averages to time integrals of correlation functions involving dissipation and the Relaxation Theorem [3] that shows how nonequilibrium systems can, under specified circumstances, relax to that quiescent state we call equilibrium. The mathematically defined dissipation function is central to each of these theorems. [4] Finally we give an extremely compact microscopic derivation of the quasi static statement of the first and second `laws' of classical thermodynamics: dU = TdS - pdV. References 1. Evans DJ and Searles DJ, The fluctuation theorem, Advances in Physics, 2002, 51,1529-1585. 2. Evans DJ, Searles DJ and Williams SR, On the fluctuation theorem for the dissipation function and its connection with response theory J. Chem. Phys. 2008, 128, 014504, ibid, 2008, 128, 249901. 3. Evans DJ, Searles DJ and Williams SR, Dissipation and the relaxation to equilibrium, J. Stat. Mech., 2009, P07029. 4. Reid JC, Williams SR, Searles DJ, Rondoni L and Evans DJ, Fluctuation relations and the foundations of statistical thermodynamics: A deterministic approach and numerical demonstration, Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond, Editors R. Klages, W. Just, C. Jarzynski, 2013 Wiley-VCH, 57-82.
Laurie Field	Conformally invariant measures on continuous loops
	In this talk we will take a tour through the family of conformally invariant measures on loops in planar domains. Starting from the Lawler-Werner Brownian loop soup and the Sheffield-Werner conformal loop ensembles (CLE), we will then outline some direct constructions that can be made using the Schramm-Loewner evolution (SLE).
Omar Foda	Conformal blocks in W_n theory
John Foxcroft
Alexandr Garbali	The Izergin--Korepin model at a root of unity
	Among the models of interacting classical statistical mechanics the Yang--Baxter (YB) integrable systems play a special role. The central model in the theory of YB integrable systems is the six vertex model. The model under our consideration is the Izergin--Korepin (IK) nineteen vertex model, which can be viewed as a generalization of the six vertex model. This model is related to the dilute loop model, physics of polymers, site percolation and has interesting conformal limit. I will talk about some recent finite size analytical developments in the study of the IK model in a certain interaction regime which is related to polymer physics.
Tim Garoni	On the coupling time distribution of the Fortuin-Kasteleyn Glauber process
	We discuss the distribution of the coupling time for the standard coupling of the Fortuin-Kasteleyn Glauber process, highlighting its relationship to the mixing time, and to the running time of the Propp-Wilson algorithm. We present a rigorous treatment in the one-dimensional case. On d-dimensional lattices, we discuss conjectures relating the asymptotic coupling time distributions to extreme value theory. Joint work with Andrea Collevecchio and Eren Elci.
Rod Gover	Calculus for conformally compact manifolds, generalising the Willmore energy, and the geometry of scale (keynote)
	The Willmore energy of a surface is a conformal measure of its failure to be conformally spherical. In physics the energy is variously called the bending energy, or rigid string action. In both physics and geometric analysis it has been the subject of great recent interest. We explain that its Euler-Lagrange equation is an extremely interesting equation in conformal geometry: the energy gradient is a fundamental curvature that is a scalar-valued hypersurface analogue of the Bach tensor (of dimension 4) of intrinsic conformal geometry. Then we show that that these surface conformal invariants, i.e. the Willmore energy and its gradient, are the lowest dimensional examples in a family of similar invariants in higher dimensions. The way these arise signals that they also have a fundamental place in conformal geometry and physics. This is joint work with Andrew Waldron arXiv:1506.02723
Jens Grimm
Tony Guttmann	Compressed random and self-avoiding walks and polygons
	We consider polymers between two impenetrable surfaces, with the surfaces being squeezed together. Polymers are modelled by two-dimensional SAWs, bridges, and SAPs. It is well known that $c_n,$ the number of $n$-step SAWs in the bulk behaves as $c_n \sim const \cdot \mu^n \cdot n^g.$ The effect of compression changes this asymptotic behaviour to include a sub-dominant stretched exponential term, so that $c_n \sim const \cdot \mu^n \cdot \mu_1^{n^\sigma} \cdot n^g'.$ We show that such behaviour is generic by explicitly solving the analogous random walk case. For SAWs, SAPs and bridges, we give results of series analysis, bolstered by probability and SLE arguments. In this way we provide heuristic arguments that the exponent $\sigma = 1/(1+d_f),$ where $d_f$ is the fractal dimension, which is $4/3$ in the case of SAWs and $2$ for random walks. Joint work with Nick Beaton, Iwan Jensen and Greg Lawler .
Nicholas Halmagyi	Local Torsional Heterotic Models for the String Landscape
	I will overview some basic aspects of the string landscape and then discuss new solutions of Heterotic torsional backgrounds on the conifold. I will discuss how these models may provide a new path for spontaneous supersymmetry breaking in string theory.
Joshua Hartigan
Md Fazlul Hoque	Direct and constructive approaches on N-dimensional superintegrable system
	Algebraic methods are powerful tools in classical and quantum mechanics. Superintegrable systems are an important class of quantum systems which can be solved using algebraic approaches. We apply algebraic approaches to obtain energy spectrum of the N-dimensional superintegrable double singular oscillators of type (n,N _ n). We develop recurrence formulas in order to generate higher order integrals of motion and the corresponding polynomial algebras of this system. We show how the su(N) symmetry algebra generated by the integrals of the N-dimensional isotropic harmonic oscillators is deformed into higher rank polynomial algebras involving Casimir operators of certain Lie algebras. We also compare and discuss the novelty of these approaches.
Kentaro Hori	Duality in quantum field theory and equivalences of categories (keynote)
	Two-dimensional (2,2) supersymmetric field theories possess the structure of two different categories, one is of algebraic geometry nature and the other is of symplectic geometry nature. There appears duality such as "mirror symmetry" and "Seiberg duality", from which one can extract mathematical conjectures concerning equivalences between important categories, like derived category of coherent sheaves, Fukaya cateogry, category of matrix factorizations, etc. Conversely, new results on such categories produce new problems in quantum field theory. I would like to describe such fruitful interaction between physics and mathematics.
E M Howard	Entanglement entropy of causal horizons
Mumtaz Hussain	An inhomogeneous wave equation and Diophantine approximation
Jesper Ipsen	Products of random matrices
Phillip Isaac	Invariants of the orthosymplectic Lie superalgebra
Jesper Jacobsen	Three-point functions in Liouville theory and conformal loop ensembles
Daniel Jafferis	Supersymmetric partitions functions of quantum field theories (keynote)
Ernie Kalnins	Separation of variables in four dimensions, the complete solution
	We discuss how separation of variables occurs in four dimensional Euclidean space. Each separation of variables has associated with it separation equations and associated constants of separation. These constants are in involution with the Hamiltonian H and they are constructed in terms of the enveloping algebra of the group E4C. This is straight forward for all types of separation except type E. We explain how to complete the calculation in this case. Joint work with W.Miller Jr. (Minnesota).
MD Khan
Jonathan Kress	An algebraic variety of superintegrable systems
Sergei Kuzenko
Diefferson Lima	Extended two site Bose-Hubbard model
Jon Links
Tianshu Liu	Towards N=2 Minimal Models
Inna Lukyanenko	Richardson-Gaudin models from the Boundary Quantum Inverse Scattering Method
	We study Richardson--Gaudin models obtained in the quasi-classical limit from the Boundary Quantum Inverse Scattering Method and investigate connections between them. In the case of diagonal reflection matrices we show that the trigonometric and rational boundary constructions are both equivalent to the trigonometric twisted-periodic construction. Thus, we don't obtain any new integrable model in this case. In the case of non-diagonal reflection matrices the situation is different. The conserved operators in the boundary construction are no longer equivalent to the ones from the twisted-periodic construction. Also, the rational and the trigonometric boundary constructions are not equivalent.
Andrzej J. Maciejewski	Application of differential Galois theory for study integrability of differential equations
Jean-Marie Maillard
Vladimir Mangazeev	On solutions of the Yang-Baxter equation related to sl(2)
	We review all known expressions for the R-matrices related to the affine algebra sl(2) and its quantum deformation. Then we look at the family of R-matrices recently derived by Chicherin, Derkachov and Spiridinov using factorization property of the L-operators. We compare their answer with our results based on the 3D approach and find explicit correspondence between two methods.
Ian Marquette	Connection between quantum systems involving the fourth Painleve transcendent and k-step rational extensions of the harmonic oscillator related to Hermite EOP
	We discuss the connection between a 1D quantum Hamiltonian involving the fourth Painleve transcendent $P_{4}$, related supersymmetric quantum mechanics, with a hierarchy of families of quantum systems called k-step rational extensions of the harmonic oscillator and related with multi-indexed $X_{m_{1},m_{2},...,m_{k}}$ Hermite EOP of type III. The connection between these exactly solvable models is established at the level of the equivalence of the Hamiltonians using rational solutions of the fourth Painleve equation in terms of generalized Hermite and Okamoto polynomials. We also relate the different ladder operators obtained by various combinations of supercharges, their zero modes and the corresponding energies.
Michael Meylan	Lax-Philips Scattering and the Break of Ice Shelves
William Moore
Jeremy Nugent	Superintegrability in pp-wave spacetimes
Judy-anne Osborn
Paul Pearce
Robert Pfeifer	Tensor network descriptions for anyonic lattice models
Sarah Post	Superintegrable systems and special functions (keynote)
	In this talk, I will discuss several new results connecting superintegrable systems and their symmetry algebras with special functions. Superintegrable systems are those which admit more integrals of motion than degrees of freedom. These additional integrals usually imply an algebraic structure making the system exactly-solvable. Because of the exact-solvability, special functions are abundant in the study of these systems. Here, I will focus on the connection with the Askey scheme of orthogonal polynomials and its connection with representations of the symmetry algebras generated by these systems. The limits of the polynomials are seen to arise naturally out of contractions of the underlying Lie algebras. Further connections with other special functions, including exceptional orthogonal polynomials, Painleve transcendents, and q-hypergeometric polynomials will also be discussed.
Maria Przybylska	Integrability of Hamiltonian systems with homogeneous potentials in flat and curved spaces
Jorgen Rasmussen
Christoph Richard	A survey of weak model sets
	Weak model sets such as the squarefree integers have recently received attention due to their connection to the M\"obius function from number theory. We survey some new results about weak model sets such as entropy bounds and dynamical properties. This is based on joint work together with Christian Huck (Bielefeld) and Gerhard Keller (Erlangen).
David Ridout
Vladimir Rittenberg	Correlation functions in conformal invariant stochastic processes
Boris Runov
Igor Samsonov	Correlation functions of conserved currents in three-dimensional superconformal theories
Johannes Schmidt	Fibonacci family of dynamical universality classes
Yibing Shen	Richardson-Gaudin form of Bethe Ansatz solution for an atomic-molecular Bose-Einstein condensate model
Yang Shi	Symmetry and geometry of discrete integrable systems
Steve Siu	Minimal Models and Free Field Realisation
	The modules of a minimal model are the space of states for a Conformal Field Theory. Classifying the representation theory of the minimal models first involves finding a specific singular vector inside a Verma module. These singular vectors are hard to compute in general. In this talk we will be focusing on the Virasoro minimal models. I will talk about how one can use free field realisation to construct this singular vector and potential generalisations to other vertex operator algebras.
Elena Tartaglia	Fused RSOS Lattice Models as Higher-Level Nonunitary Minimal Cosets
	It is well known that the continuum scaling limit of the critical Forrester-Baxter Restricted-Solidon-Solid (RSOS) models, with crossing parameter $\lambda = (m'-m)\pi/m'$, is described by the nonunitary minimal models $\mathcal{M}(m,m')$. More generally, we argue that at criticality the $n\times n$ fused RSOS models with relate to the higher-level non-unitary coset minimal models. Specifically, starting with the off-critical elliptic models for $n=1,2,3$, we calculate the Corner Transfer Matrix one-dimensional sums to obtain finitised branching functions and infer a general formula for the central charges. Extending the work of Schilling, we also conjecture finitized bosonic branching functions for general $n$ and check that these agree with the one-dimensional sums for $n=1,2,3$ out to system size $N=14$. In this way we identify the continuum scaling limit of $n\times n$ fused RSOS models with known level-$n$ nonunitary coset theories.
Naghmana Tehseen	Frobenius integrability of evolution equations and travelling wave solutions
	The integrability of some evolution equations in discussed in terms of the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. In particular, we give a powerful result which explains the extraordinary integrability of some of these equations. We also discuss "local" conservations laws for evolution equations in general and demonstrate all the results for the Kortewegñde Vries equation.
Matthew Ussher	A Pattern Calculus for Lie Superalgebras
	A pattern calculus is a technique for explicitly calculating and manipulation reduced Wigner coefficients. I'll consider how to construct a pattern calculus for Lie superalgbras by considering how we do it for Lie algebras in the context of U(n).
Peter Vassiliou	Symmetry Reduction, Contact Geometry and Nonlinear Control
Michael Wheeler	A new way of calculating Hall polynomials
	Hall polynomials are the structure constants of the Hall algebra, and constitute a one-parameter generalization of the famous Littlewood--Richardson coefficients. Using an integrable model of deformed bosons, I will present a new combinatorial expression for these polynomials in terms of certain (intersecting) lattice paths.
Simon Wood	Minimal models from free fields
Yao-Zhong Zhang	On the k-photon quantum Rabi model

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