Presentation Details 

Abrahams, I David 
Corner singularities and improved convergence of eigenfunction expansions in acoustics, electromagnetics and elasticity 
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It is well known that the solution of waveguide problems in acoustics, electromagnetism, and elasticity are expressible as an infinite sum of discrete modes (or eigenfunctions). A variety of methods, such as Fourier transforms or mode matching, may be employed to determine the scattered field induced by waves impinging on a geometric discontinuity, edge or other inhomogeneity, and these approaches often reduce to finding the modal coefficients from an infinite linear system of equations. Unfortunately, it is well known that in many cases the convergence of such systems is very slow, so that very large truncated systems have to be taken to ensure an accurate solution.
Many methods have been proposed to improve convergence of linear systems; however, this talk will demonstrate, by way of several specific problems, how an a priori understanding of the physical solution close to corners or other irregularities can be used be used to dramatically improve the convergence. The procedure is undertaken in complex wavenumber space. 
Blyth, Mark 
Deformation of an elastic cell under inviscid flow 
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We use complex variable methods to study the nonlinear deformation of a twodimensional elastic cell placed into a uniform flow. The flow is assumed to be inviscid and irrotational. The problem is governed by two dimensionless parameters which reflect the transmural pressure difference between the interior of the cell and the ambient pressure at infinity, and the strength of the oncoming flow. In the absence of flow, and for sufficiently low transmural pressure across the cell wall, the cell is circular; Flaherty et al (1972) showed that as the transmural pressure increases through different threshold values, buckled states with mfold rotational symmetry come into play, and that each of these eventually exhibits a point of self contact at a critical pressure. We focus on the effect of flow on the cell and contrast with the behaviour found for a twodimensional bubble in a uniform stream. In the second part of the talk we consider the additional effect of circulation around the cell as a simple model of a deformable aerofoil experiencing lift. 
Dallaston, Michael 
Asymptotic selection of selfsimilar rupture solutions to a generalised thin film equation 
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This talk concerns similarity solutions to a thin film equation $h_t + (h^mh_{xxx} + h^nh_x)_x=0$, in which the exponents $m$ and $n$ are arbitrary. Such a model can describe thin film phenomena on different scales, such as rupture due to van der Waals force, destabilisation due to gravity (RayleighTaylor instability), or thermocapillary forces, depending on the exponent $n$ (generally $m=3$ for a film on a solid substrate).
Computations show that there are a countably infinite number of similarity solutions, which merge via saddlenode bifurcations as $n$ is changed. In this talk we will discuss the asymptotic selection of these solution branches out of a continuum using exponential asymptotic techniques, and the mechanism by which solution branches merge. This extends on a work by Chapman, Trinh & Witelski [SIAM J Appl Math, 2013 (73):232253], who performed the asymptotic analysis for the special case relevant to van der Waals rupture. 
Deaño, Alfredo 
Special function solutions of Painlevé II and IV: asymptotic and numerical study 
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Special function solutions of the Painlevé equations are important in the theory of orthogonal polynomials, integrable systems and in the study of random matrices. These solutions of Painlevé equations show remarkable structure, since they can be constructed as Wronskian determinants involving a certain seed function, which is a combination of Airy or Weber functions for PII and PIV respectively. In this talk, we will address some asymptotic properties of these functions in the complex plane, as well as their pole fields and some computational challenges. 
GómezUllate Oteiza, David 
Durfee rectangles, exceptional Hermite polynomials and rational solutions to Painlevé equations 
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We will introduce certain symmetries of Wronskian determinants whose entries are Hermite polynomials, which have a nice combinatorial interpretation in terms of Maya diagrams and Durfee rectangles. Some applications to exceptional Hermite polynomials and rational solutions to Painlevé equations will be also discussed. 
Grava, Tamara 
Universality of critical behaviour in Hamiltonian PDEs 
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I study and classify various type of critical behaviours to solutions of Hamiltonian Partial Differential Equations (PDEs) in one and two spatial dimensions, that have their origin in optics and fluid dynamic. Painleve' equations and special functions play an important role in the description of these critical behaviours. 
Halburd, Rod 
Integrable delaydifferential equations 
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Several delaydifferential equations with limits to the usual (differential) Painlevé equations will be obtained (a) as the reductions of integrable differentialdifference equations, (b) from Bäcklund transformations of Painlevé equations, and (c) by looking for delaydifferential equations admitting meromorphic solutions with regular value distribution in the complex plane. Delaydifferential analogues of special cases of the QuispelRobertsThompson (QRT) mapping will also be described. 
Himonas, Alex 
The unified transform method and wellposedness of nonlinear dispersive equations 
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The unified transform method (UTM), which is also known as the Fokas transform method, was introduced in late nineties as the analogue of the inverse scattering transform machinery for integrable nonlinear equations on the halfline. It was later understood that it also has significant implications for linear initialboundary value problems. In this talk, this method is employed in a new direction, namely for showing wellposedness of nonlinear dispersive equations, including the nonlinear Schr"odinger equation and the Kortewegde Vries equation, on the halfline with data in Sobolev spaces. 
Hitzazis, Iasonas 
Linear elliptic PDEs in a cylindrical domain with a polygonal crosssection 
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Integral representations for the solution of the Laplace, modified Helmholtz, and Helmholtz equations can be obtained using Green’s theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a wellposed boundary value problem (BVP) one of these functions is unknown. A new transform method for solving BVPs for linear and for integrable nonlinear partial differential equations (PDEs), usually referred to as the unified transform or the Fokas method, was introduced in the late nineties. For linear elliptic PDEs in two dimensions, this method first, by employing two algebraic equations formulated in the Fourier plane, provides an elegant approach for determining the Dirichlet to Neumann map, i.e., for constructing the unknown boundary values in terms of the given boundary data. Second, this method constructs novel integral representations of the solution in terms of integrals formulated in the complex Fourier plane. In the present paper, we extend this novel approach to the case of the Laplace, modified
Helmholtz, and Helmholtz equations, formulated in a threedimensional cylindrical domain with a polygonal crosssection. This is joint work with A. S. Fokas. 
Johnson, Edward 
Rotating vortical outflows 
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A simple, fullynonlinear, dispersive, quasigeostrophic model is put forward to isolate the vorticity dynamics of coastal outflows. The model is sufficiently simple so as to allow highly accurate, numerical integration of the full problem and also explicit, fullynonlinear solutions for the evolution of a uniform velocity outflow in the hydraulic limit. The flow evolution depends strongly on the sign of the vorticity of the expelled fluid and on the ratio of the internal Rossby radius to the vortexsource scale, $V_0/D^2Pi_0^{1/2}$, of the flow, where $D$ measures the outflow depth, $Pi_0$, the perturbation potential vorticity in the outflow and $V_0$ the volume flux of the outflow. Comparison of the explicit hydraulic solutions with the numerical integrations shows that the analytical solutions predict the flow development well with differences ascribable to dispersive Rossby waves on the current boundary and changes in the source region captured by the full equations but not present in the hydraulic solutions. 
Kisil, Anastasia 
Approximate matrix WienerHopf factorisations and applications to problems in acoustics 
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First, I will introduce the WienerHopf method which extends the separation of variables technique (in Cartesian coordinate) used to investigate PDEs. It provided analytic and systematic methodology for previously unapproachable problems. One of the problems discussed will be scattering of a sound wave by an infinite periodic grating composed of rigid plates (joint work with I. D. Abrahams).
I will also talk about a matrix WienerHopf problems which is motivated by studying the effect of a finite elastic trailing edge on noise production (joint work with N. Peake, L. Ayton). The approximate factorisation of this matrix with exponential phase factors is achieved using an iterative procedure which makes use of the scalar WienerHopf problem arising for each junction. 
Llewellyn Smith, Stefan 
Solving matrix WienerHopf problems numerically via RiemannHilbert problems 
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A RiemannHilbert (RH) problem asks for the construction of a function that is analytic everywhere in the complex plane except along a given curve where it has a prescribed jump. The WienerHopf (WH) method was develoepd to solve mixed boundaryvalue problem; its fundamental essence consists in factoring a function into parts that are analytic in different domains. Explicit formulas to compute this factorization for the scalar case, but no general method is know for matrix problems. The relation between RH problems and the WH method has been known for a long time. Numerical methods to solve RH problems have been the subject of considerable recent work. We examine their use in solving WienerHopf problems, in particular matrix problems. We examine a number of model problems and focus on obtaining numerical results that can be used to compute quantities such as the farfield amplitude and other properties of the physical solution. It turns out that particular attention needs to be paid to the decay properties of the relevant functions. 
Lombardo, Sara 
Linear stability analysis of integrable partial differential equations 
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Analytical methods of the theory of integrable partial differential equations (PDEs) in $1 + 1$ dimensions have been successfully applied to investigate a number of wave propagation models of physical interest. This talk shows how to address the issue of linear stability of wave solutions by means of these methods. By imitating the standard steps followed when dealing with non integrable equations, we show how the linear stability of solutions of integrable PDEs can be effectively analysed by using their Lax representation. The most relevant application of this scheme is the analysis of the background continuous wave solution. The talk is based on work done in collaboration with Antonio Degasperis, University of Rome La Sapienza, Rome, Italy and Matteo Sommacal, Northumbria University, Newcastle upon Tyne, UK. 
Louca, Elena 
A new transform approach to biharmonic boundary value problems in polygonal and circular domains 
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Motivated by modelling challenges arising in microfluidics and lowReynoldsnumber swimming, we present a new transform approach for solving biharmonic boundary value problems in twodimensional polygonal and circular domains. The method is an extension of earlier work by Crowdy & Fokas [Proc. Roy. Soc. A, 460, (2004)] and provides a unified general approach to finding quasianalytical solutions to a wide range of problems in lowReynoldsnumber hydrodynamics and plane elasticity. 
Loureiro, Ana 
On starsymmetric polynomials with a classical behaviour 
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I will discuss sequences of polynomials of a single variable that are orthogonal with respect to a vector of weights defined in the complex plane. Such polynomial sequences satisfy a recurrence relation of finite (and fixed) order higher than 2. They share a number of properties that mimic those of standard orthogonality on $L_2$ spaces. The main focus will be on polynomial sequences possessing a threestar symmetry and whose multiple orthogonality is preserved under the action of the derivative operator. 
Mansfield, Elizabeth 
Discrete moving frames and discrete variational problems 
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To follow 
MineevWeinstein, Mark 
Thermodynamics of the Laplacian growth 
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The methods of equilibrium statistical thermodynamics are applied to Laplacian growth by using its remarkable connection with a random matrix theory. The Laplacian growth equation is obtained from the variation principle and describes adiabatic (quasistatic) thermodynamic processes in the twodimensional Dyson gas. By using Einstein's theory of thermodynamic fluctuations we consider transitional probabilities between thermodynamic states, which are in a onetoone correspondence with planar domains. Transitions between these domains are described by the stochastic Laplacian growth equation, while the transitional probabilities coincide with the freeparticle propagator on the infinite dimensional complex manifold with the Kahler metric. 
Olver, Peter 
Dispersive quantization of linear and nonlinear waves 
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The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, nondifferentiable profiles at irrational times. Similar phenomena have been observed in optics and quantum mechanics, and lead to intriguing connections with exponential sums arising in number theory. Ramifications and recent progress on the analysis, numerics, and extensions to nonlinear wave models will be discussed. 
Protas, Bartosz 
On the stability of freeboundary problems: a case study in vortex dynamics 
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Many problems in science and engineering are described in terms of equilibrium shapes on which certain conditions are imposed and which separate regions where the solution may have different properties. A prototypical problem of this type involves 2D inviscid vortex equilibria where constantvorticity vortex patches are embedded in a potential flow. Methods of complex analysis offer a particularly elegant and efficient description of such systems. Studying linear stability of such freeboundary problems is however challenging as it requires characterization of the sensitivity of the equilibrium solutions with respect to suitable perturbations of the boundary. We will demonstrate that such questions can be in fact systematically addressed using techniques of "shape calculus" applied to formulations based on singular integral equations. In the context of vortex dynamics we use this approach to obtain an equation characterizing the stability of general equilibrium solutions involving vortex patches. Certain classical results of vortex stability are then derived as special cases. Finally, this approach is employed to solve an open problem concerning the linear stability of Hill's vortex to 3D axisymmetric perturbations, which leads to some unexpected findings. 
Sheils, Natalie 
The heat equation with imperfect thermal contact in a composite medium 
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The problem of heat conduction in onedimensional piecewise homogeneous composite materials is examined by providing an explicit solution of the onedimensional heat equation in each domain. The location of the interfaces is known, but neither temperature nor heat flux are prescribed there. We find a solution using the Unified Transform Method, due to Fokas and collaborators, applied to interface problems and compute solutions numerically. 
Smith, David 
Nonlocal problems for linear evolution equations 
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Linear evolution equations, such as the heat equation, are commonly studied on finite spatial domains via initialboundary value problems. In place of the boundary conditions, we consider ''multipoint conditions'', where one specifies some linear combination of the solution and its derivative evaluated at internal points of the spatial domain, and ''nonlocal'' specification of the integral over space of the solution against some continuous weight. 
Trichtchenko, Olga 
Solutions and stability for flexuralgravity waves 
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We will describe the methods used to compute solutions to waves under a sheet of ice and show the stability results in different asymptotic regions for these solutions. 
Vasconcelos, Giovani 
Timedependent solutions and the selection problem for multiple bubbles in a HeleShaw cell 
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The unsteady motion of a finite assembly of bubbles in a HeleShaw channel is analysed in the case when surface tension is neglected. A general exact solution for the problem is obtained in terms of a conformal map from a multiply connected circular domain to the fluid region exterior to the bubbles. The corresponding mapping function is given explicitly in terms of certain special transcendental functions known as the secondary SchottkyKlein prime functions. The timedependent parameters of the mapping are written in terms of certain conserved quantities associated with the singularities of Schwarz function of the interface and its behaviour at infinity. These equations are solved numerically for the parameters and explicit examples of timeevolving multibubble configurations are presented. It is shown that for a very general class of initial conditions, for which the solutions exist for all times, the solutions approach (in the long time limit) a steadily moving configuration where the bubbles move with speed U=2 relative to the background flow. This confirms the conjecture put forward by the author and MineevWeinstein (Phys. Rev. E 89, 061003, 2014) that velocity selection in a HeleShaw cell does not require surface tension, with U=2 being the generally selected velocity whether it is for a finger, a single bubble or multibubble configurations. 