The complex analysis toolbox: new techniques and perspectives
Monday 9th September 2019 to Friday 13th September 2019
09:20 to 09:50  Registration  
09:50 to 10:00  Welcome from David Abrahams (Isaac Newton Institute)  
10:00 to 11:00 
Mark Ablowitz New Integrable Nonlocal Nonlinear Equations and Solitons
Solitons and the Inverse Scattering Transform (IST) are well known in the Math/Physics community. Motivated by recent results in physics involving PT symmetry a surprisingly large number of `simple’ integrable nonlocal wave equations have been identified; their solutions, including solitons and properties will be discussed. The method, IST, requires substantial complex analysis. The nonlocal nonlinear Schrodinger equations arise universally; they are related to complex solutions of the KortewegdeVries, nonlinear KleinGordon and water wave equations. 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Beatrice Pelloni Boundary value problems on a finite interval, fractalisation and revivals
I will describe the behaviour of equations posed on a finite interval, and in particular the “Talbot effect”, a phenomenon known in optics and quantum mechanics, studied by M. Berry in the 1990s and rediscovered in the context of dispersive equations by Peter Olver in recent years. In this context, this effect implies that the solution of periodic problems exhibits either revivals of the initial condition, or fractalisation. To study the extent of this effect, we use the solution representation obtained by the Unified Transform of Fokas, and numerical experimentation. This is joint work with David Smith, Lyonell Boulton and George Farmakis. 
INI 1  
12:30 to 14:00  Lunch at Westminster College  
14:00 to 15:00 
Jonatan Lenells Large gap asymptotics at the hard edge for MuttalibBorodin ensembles
I will present joint work with Christophe Charlier and Julian Mauersberger. We consider the limiting process that arises at the hard edge of MuttalibBorodin ensembles. This point process depends on $\theta > 0$ and has a kernel built out of Wright's generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the form \begin{equation*} \mathbb{P}(\mbox{gap on } [0,s]) = C \exp \left( a s^{2\rho} + b s^{\rho} + c \ln s \right) (1 + o(1)) \qquad \mbox{as }s \to + \infty, \end{equation*} where the constants $\rho$, $a$, and $b$ have been derived explicitly via a differential identity in $s$ and the analysis of a RiemannHilbert problem. Their method can be used to evaluate $c$ (with more efforts), but does not allow for the evaluation of $C$. In this work, we obtain expressions for the constants $c$ and $C$ by employing a differential identity in $\theta$. When $\theta$ is rational, we find that $C$ can be expressed in terms of Barnes' $G$function. We also show that the asymptotic formula can be extended to all orders in $s$. 
INI 1  
15:00 to 15:30 
Andre Nachbin Conformally mapping water waves: top, bottom or sides.
I will present a brief overview of recent work showcasing conformal mapping's important role on surface waterwave dynamics. Conformal mapping can be used to flatten the free surface or a highly irregular bottom topography. It has also been used along the sides of forked channel regions, leading to a Boussinesq system with solitary waves on a graph. Mapping a highly variable bottom topography, among other features, allows the construction of a DirichlettoNeumann operator over a polygonal bottom profile. One very recent example applies to a hydrodynamic pilotwave model, capturing two bouncing droplets confined in cavities, where they can synchronize as nonlinearly coupled oscillators. Finally, on another topic, I will briefly present a very recent result displaying a spectrally accurate finite difference operator. This difference operator is constructed by unconventional means, having in mind complex analytic functions.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Alfredo Deaño Painlevé equations and nonHermitian random matrix ensembles
In this talk we present recent results on the connection between Painlevé equations and NxN nonHermitian ensembles of random matrices, in particular those models arising from classical cases with the addition of charges in the complex plane. The link with Painlevé transcendents can be established both for finite N and as the size of the matrices N tends to infinity, involving different families of solutions in each case. As examples we consider the lemniscate ensemble and truncations of unitary matrices. This is joint work with Nick Simm (University of Sussex, United Kingdom). 
INI 1  
17:00 to 18:00  Welcome Wine Reception at INI 
09:00 to 10:00 
Peter Clarkson Painleve Equations  Nonlinear Special Functions I
The six Painleve equations, whose solutions are called the Painleve transcendents, were derived by Painleve and his colleagues in the late 19th and early 20th centuries in a classification of second order ordinary differential equations whose solutions have no movable critical points. In the 18th and 19th centuries, the classical special functions such as Bessel, Airy, Legendre and hypergeometric functions, were recognized and developed in response to the problems of the day in electromagnetism, acoustics, hydrodynamics, elasticity and many other areas. Around the middle of the 20th century, as science and engineering continued to expand in new directions, a new class of functions, the Painleve functions, started to appear in applications. The list of problems now known to be described by the Painleve equations is large, varied and expanding rapidly. The list includes, at one end, the scattering of neutrons off heavy nuclei, and at the other, the distribution of the zeros of the Riemannzeta function on the critical line Re(z) =1/2. Amongst many others, there is random matrix theory, the asymptotic theory of orthogonal polynomials, selfsimilar solutions of integrable equations, combinatorial problems such as the longest increasing subsequence problem, tiling problems, multivariate statistics in the important asymptotic regime where the number of variables and the number of samples are comparable and large, and also random growth problems. The Painleve equations possess a plethora of interesting properties including a Hamiltonian structure and associated isomonodromy problems, which express the Painleve equations as the compatibility condition of two linear systems. Solutions of the Painleve equations have some interesting asymptotics which are useful in applications. They possess hierarchies of rational solutions and oneparameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further the Painleve equations admit symmetries under affine Weyl groups which are related to the associated Backlund transformations. In these lectures I shall first review many of the remarkable properties which the Painleve equations possess. In particular I will discuss rational solutions of Painleve equations. Although the general solutions of the six Painleve equations are transcendental, all except the first Painleve equation possess rational solutions for certain values of the parameters. These solutions are usually expressed in terms of logarithmic derivatives of special polynomials that are Wronskians, often of classical orthogonal polynomials such as Hermite and Laguerre. It is also known that the roots of these special polynomials are highly symmetric in the complex plane. The polynomials arise in applications such as random matrix theory, vortex dynamics, in supersymmetric quantum mechanics, as coefficients of recurrence relations for semiclassical orthogonal polynomials and are examples of exceptional orthogonal polynomials. 
INI 1  
10:00 to 11:00 
Walter Van Assche Zero distribution of discrete orthogonal polynomials on a qlattice
We investigate the asymptotic distribution of the zeros of orthogonal polynomials $P_n$ for which the orthogonality measure is supported on the $q$lattice $\{q^k,k=0,1,2,\ldots\}$, where $0

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Arno Kuijlaars The spherical model with two external charges.
I will discuss a model from electrostatics where N charged particles with charge 1/N distribute themselves over the unit sphere in the presence of a finite number of fixed charges. In the large N limit the particles concentrate on a part of the sphere with uniform density. This is the droplet. After stereographic projection, this model is analyzed in the complex plane with tools from logarithmic potential theory. For the case of two equal charges we compute a mother body, which is then used to determine the droplet explicitly. This is joint work Juan Criado del Rey. 
INI 1  
12:30 to 14:00  Lunch at Westminster College  
14:00 to 15:00 
Kerstin Jordaan Asymptotic zero distribution of generalized hypergeometric polynomials
In this talk I will present a brief overview of techniques used to determine the zero distribution of various classes of generalized hypergeometric polynomials as the degree tends to infinity with special consideration given to a class of Gauss hypergeometric polynomials, known as PseudoJacobi polynomials, that are closely connected to Jacobi polynomials with complex parameters and purely imaginary argument.

INI 1  
15:00 to 15:30 
Jan Zur Computing all zeros of harmonic mappings in the plane
We present a continuation method to compute all zeros of certain harmonic mappings $f$ in the complex plane. While tracing the homotopy curves of $f$ is done by a prediction correction approach, the main difficulty is to handle the bifurcations and turning points. To achieve this we study the critical curves and caustics of $f$. Moreover, we illustrate our method with several examples and discuss possible extensions. This is joint work with Olivier Sète (TU Berlin). 
INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Davide Guzzetti A technique to solve some isomonodromy deformation problems, with applications
Since the workshop focuses on new techniques and mathematical ideas within the area of complex analysis over the past few years, I will present an extension of the isomonodromy deformation theory which allows, in some nongeneric cases, to perform explicit computations of monodromy data and fundamental solutions of isomonodromic systems in a relatively simple way. I will present applications to Painleve' equations and Frobenius manifolds (particularly quantum cohomology), based on joint works with G. Cotti and B. Dubrovin.

INI 1 
09:00 to 10:00 
Peter Clarkson Painleve Equations  Nonlinear Special Functions II
The six Painleve equations, whose solutions are called the Painleve transcendents, were derived by Painleve and his colleagues in the late 19th and early 20th centuries in a classification of second order ordinary differential equations whose solutions have no movable critical points. In the 18th and 19th centuries, the classical special functions such as Bessel, Airy, Legendre and hypergeometric functions, were recognized and developed in response to the problems of the day in electromagnetism, acoustics, hydrodynamics, elasticity and many other areas. Around the middle of the 20th century, as science and engineering continued to expand in new directions, a new class of functions, the Painleve functions, started to appear in applications. The list of problems now known to be described by the Painleve equations is large, varied and expanding rapidly. The list includes, at one end, the scattering of neutrons off heavy nuclei, and at the other, the distribution of the zeros of the Riemannzeta function on the critical line Re(z) =1/2. Amongst many others, there is random matrix theory, the asymptotic theory of orthogonal polynomials, selfsimilar solutions of integrable equations, combinatorial problems such as the longest increasing subsequence problem, tiling problems, multivariate statistics in the important asymptotic regime where the number of variables and the number of samples are comparable and large, and also random growth problems. The Painleve equations possess a plethora of interesting properties including a Hamiltonian structure and associated isomonodromy problems, which express the Painleve equations as the compatibility condition of two linear systems. Solutions of the Painleve equations have some interesting asymptotics which are useful in applications. They possess hierarchies of rational solutions and oneparameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further the Painleve equations admit symmetries under affine Weyl groups which are related to the associated Backlund transformations. In these lectures I shall first review many of the remarkable properties which the Painleve equations possess. In particular I will discuss rational solutions of Painleve equations. Although the general solutions of the six Painleve equations are transcendental, all except the first Painleve equation possess rational solutions for certain values of the parameters. These solutions are usually expressed in terms of logarithmic derivatives of special polynomials that are Wronskians, often of classical orthogonal polynomials such as Hermite and Laguerre. It is also known that the roots of these special polynomials are highly symmetric in the complex plane. The polynomials arise in applications such as random matrix theory, vortex dynamics, in supersymmetric quantum mechanics, as coefficients of recurrence relations for semiclassical orthogonal polynomials and are examples of exceptional orthogonal polynomials. 
INI 1  
10:00 to 11:00 
Bruno Carneiro da Cunha Isomonodromic tau functions, the constructive approach to conformal maps, and black holes.
Recent developments on the relation between the RiemannHilbert problem and the representation theory of Virasoro algebras allowed for explicit expansions of the isomonodromic tau functions in terms of conformal blocks. In this talk I will describe how these expansions can be used to constructively solve the connection problem of ordinary differential equations of the Fuchsian type. The simplest nontrivial case of 4 regular singular points (the Heun equation)  as well as a particular confluent limit  are solved by generic Painlevé transcendents of the sixth and fifth type. On the formal side, these relations allow us to conjecture an interpretation of the zeros of the tau functions in the general case. On the application side, the explicit expansions are useful for high precision numerical calculations of the accessory parameters of conformal maps, as well as the determination of (quasi)normal modes of metric vibrations for a variety of black hole backgrounds in general relativity. Coauthors include: T. Anselmo, J.J. BarragánAmado, J. P. Cavalcante, R. Nelson, D. Crowdy and E. Pallante. 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Oleg Lisovyy Painlevé functions, Fredholm determinants and combinatorics
I will explain how to associate a tau function to the RiemannHilbert problem set on a union of nonintersecting smooth closed curves with generic jump matrix. The main focus will be on the onecircle case, relevant to the analysis of Painlevé VI equation and its degenerations to Painlevé V and III. The tau functions in question will be defined as block Fredholm determinants of integral operators with integrable kernels. They can be alternatively represented as combinatorial sums over tuples of Young diagrams which coincide with the dual NekrasovOkounkov instanton partition functions for RiemannHilbert problems of isomonodromic origin.

INI 1  
12:30 to 14:00  Lunch at Westminster College  
14:00 to 18:00  Free afternoon  
19:30 to 22:00 
Formal Dinner at Emmanuel College 
09:00 to 10:00 
Peter Clarkson Painleve Equations  Nonlinear Special Functions III
The six Painleve equations, whose solutions are called the Painleve transcendents, were derived by Painleve and his colleagues in the late 19th and early 20th centuries in a classification of second order ordinary differential equations whose solutions have no movable critical points. In the 18th and 19th centuries, the classical special functions such as Bessel, Airy, Legendre and hypergeometric functions, were recognized and developed in response to the problems of the day in electromagnetism, acoustics, hydrodynamics, elasticity and many other areas. Around the middle of the 20th century, as science and engineering continued to expand in new directions, a new class of functions, the Painleve functions, started to appear in applications. The list of problems now known to be described by the Painleve equations is large, varied and expanding rapidly. The list includes, at one end, the scattering of neutrons off heavy nuclei, and at the other, the distribution of the zeros of the Riemannzeta function on the critical line Re(z) =1/2. Amongst many others, there is random matrix theory, the asymptotic theory of orthogonal polynomials, selfsimilar solutions of integrable equations, combinatorial problems such as the longest increasing subsequence problem, tiling problems, multivariate statistics in the important asymptotic regime where the number of variables and the number of samples are comparable and large, and also random growth problems. The Painleve equations possess a plethora of interesting properties including a Hamiltonian structure and associated isomonodromy problems, which express the Painleve equations as the compatibility condition of two linear systems. Solutions of the Painleve equations have some interesting asymptotics which are useful in applications. They possess hierarchies of rational solutions and oneparameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further the Painleve equations admit symmetries under affine Weyl groups which are related to the associated Backlund transformations. In these lectures I shall first review many of the remarkable properties which the Painleve equations possess. In particular I will discuss rational solutions of Painleve equations. Although the general solutions of the six Painleve equations are transcendental, all except the first Painleve equation possess rational solutions for certain values of the parameters. These solutions are usually expressed in terms of logarithmic derivatives of special polynomials that are Wronskians, often of classical orthogonal polynomials such as Hermite and Laguerre. It is also known that the roots of these special polynomials are highly symmetric in the complex plane. The polynomials arise in applications such as random matrix theory, vortex dynamics, in supersymmetric quantum mechanics, as coefficients of recurrence relations for semiclassical orthogonal polynomials and are examples of exceptional orthogonal polynomials. 
INI 1  
10:00 to 11:00 
Björn Gustafsson Geometric function theory and vortex motion: the role of connections
We discuss point vortex dynamics on a closed twodimensional
Riemann manifolds from the point of view of affine and other connections.
The speed of a vortex then comes out as the difference between two
affine connections, one derived from the coordinate Robin function and the
other being the LeviCivita connection associated to the Riemannian metric.
In a Hamiltonian formulation of the vortex dynamics, the Hamiltonian function consists of two main terms. One of them is a quadratic form based on a matrix whose entries are Green and Robin functions, while the other describes the energy contribution from those circulating flows besides those which are implicit in the Green functions. These two terms are not independent of each other, and one major issue is trying to understand the exchange of energy between them. 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Elena Luca Viscous flows in channel geometries
Motivated by modelling challenges arising in microfluidics and lowReynoldsnumber swimming, we consider viscous flows in twodimensional channels and present new transform methods for analysing such problems. The new methods provide a unified general approach to finding quasianalytical solutions to a wide range of problems in lowReynoldsnumber hydrodynamics and plane elasticity. In this talk, we focus on pressuredriven flows in channel geometries with linear expansions and angled transitions. This is joint work with Prof. Stefan G. Llewellyn Smith (UCSD).

INI 1  
12:30 to 14:00  Lunch at Westminster College  
14:00 to 15:00 
Mihai Putinar Hyponormal quantization of planar domains
By replacing the identity operator in Heisenberg commutation relation [T*,T]=I by a rankone projection one unveils an accessible spectral analysis classification with singular integrals of Cauchy type as generic examples. An inverse spectral problem for this class of (hyponormal) operators can be invoked for encoding and decoding (partial) data of 2D pictures carrying a grey shade function. An exponential transform, the two dimensional analog of a similar operation on Cauchy integrals introduced by A, Markov in his pioneering work on 1D moment problems, provides an effective dictionary between "pictures" in the frequency domain and "matrices" in the state space interpretation. A natural RiemannHilbert problem lies at the origin of this kernel with potential theoretic flavor. Quadrature domains for analytic functions are singled out by a rationality property of the exponential transform, and hence an exact reconstruction algorithm for this class of black and white shapes emerges. A two variable diagonal Pade approximation scheme and some related complex orthogonal polynomials enter into the picture, with their elusive zero asymptotics. Most of the results streaming from two decades of joint work with Bjorn Gustafsson. 
INI 1  
15:00 to 15:30 
Vikas Krishnamurthy Steady point vortices in a field of Stuarttype vorticity
A
new family of exact solutions to the twodimensional steady incompressible
Euler equation is presented. The solutions comprise two point vortices of unit
circulation – a point vortex pair – embedded in a smooth sea of nonzero
vorticity of “Stuarttype”. The solution is one of the simplest examples of a
rich and diverse array of similar global equilibria of the Euler equation
identified by the authors. We also examine the point vortex limit of these new
Stuartembedded point vortex equilibria which results in a tworealparameter
family of smoothly deformable asymmetric point vortex equilibria in an
otherwise irrotational flow.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 16:30 
Dmitry Ponomarev Kelvin transform and Fourier analysis for explicit reconstruction formulae in paleomagnetic context
We consider socalled inverse magnetization problem in paleomagnetic context. In such a problem the aim is to recover the average remaneWe consider socalled inverse magnetization problem in the paleomagnetic context. In such a problem the aim is to recover the average remanent magnetization of a sample from measurements of one component of magnetic field in a planar region above the sample. To achieve this goal, two methods based on complexanalysis and harmonic function theory were specially developed. The first is based on Kelvin transformation mapping planar data to the family of spheres which is then followed by asymptotical analysis of spherical harmonics projection integrals. The second method is due to direct twodimensional Fourier analysis of the data in a suitable neighborhood of the origin. The latter becomes possible after a suitable asymptotic completion of the original measurement data has been performed. The obtained explicit formulas estimating net moment components in terms of the normal component of the measured magnetic field show good agreement with synthetically generated numerical and experimental data on samples with fairly localized magnetization distributions. It is an interesting example how the problem can be solved using tools of discrete and continuous harmonic analysis. The talk is based on a joint work with Laurent Baratchart, Juliette Leblond (INRIA Sophia Antipolis, France) and Eduardo Andrade Lima (MIT, USA). 
INI 1  
16:30 to 17:00 
Nathan Hayford A Baker Function for Laplacian Growth and Phase Transitions
Laplacian growth describes the evolution of an
incompressible fluid droplet with zero surface tension in 2D, as fluid is
pumped through a well into the droplet. A major obstacle in the theory of
Laplacian growth is the formation of finitetime singularities (cusps) that
form on the boundary of the fluid droplet. Although some work has been done
with regards to continuation of the solution past this critical point, most
results are phenomenological in nature, and a general theory is yet to be
developed. Due to Laplacian growth's realization as a dispersionless
limit of the 2D Toda Hierarchy, we investigate certain scaling limits of this
hierarchy's Baker function. We pose the question, "what can the
Baker function tell us about phase transitions in the droplet?", for
particular classes of initial domains.

INI 1 
09:00 to 10:00 
Mohamed Nasser PlgCirMap: A MATLAB toolbox for computing the conformal maps from polygonal multiply connected domains onto circular domains
In [1], the author has presented a method for computing the conformal mapping form a given bounded or unbounded multiply connected domains onto circular domain. The method is based on a fast numerical implementation of Koebe's iterative method using the boundary integral equation with the generalized Neumann kernel which can be solved fast and accurately with the help of FMM [2]. The method gives accurate results even when the given domain is a polygonal domain.
In this talk, the method presented in [1] will be used to develop a MATLAB toolbox for computing the conformal mapping $w=f(z)$ from a given polygonal multiply connected domain $G$ onto a circular domain $D$ and its inverse $z=f^{1}(w)$. The boundaries of the polygons are assumed to be piecewise smooth Jordan curves without cusps. The toolbox can be used even for domains with high connectivity.
References.
[1] M.M.S. Nasser, Fast computation of the circular map, Comput. Methods Funct. Theory 15 (2) (2015) 187223.
[2] M.M.S. Nasser, Fast solution of boundary integral equations with the generalized Neumann kernel, Electron. Trans. Numer. Anal. 44 (2015) 189229.

INI 1  
10:00 to 11:00 
Lesley Ward The harmonicmeasure distribution function of a planar domain, and the SchottkyKlein prime function
The $h$function or harmonicmeasure distribution function $h(r) = h_{\Omega, z_0}(r)$ of a planar region $\Omega$ with respect to a basepoint $z_0$ in $\Omega$ records the probability that a Brownian particle released from $z_0$ first exits $\Omega$ within distance $r$ of $z_0$, for $r > 0$. For simply connected domains $\Omega$ the theory of $h$functions is now well developed, and in particular the $h$function can often be computed explicitly, making use of the Riemann mapping theorem. However, for multiply connected domains the theory of $h$functions has been almost entirely out of reach. I will describe recent work showing how the SchottkyKlein prime function $\omega(\zeta,\alpha)$ allows us to compute the $h$function explicitly, for a model class of multiply connected domains. This is joint work with Darren Crowdy, Christopher Green, and Marie Snipes.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Rod Halburd Local and global branching of solutions of differential equations
We will consider differential equations with movable branch points
in the complex domain. We will describe
families of equations for which we can prove that the only movable
singularities of solutions are algebraic.
In general the global structure of these solutions is very complicated,
despite the fact that locally all branching is finite. We will show how to determine all equations
within particular families for which the solutions are globally finitely
branched. These equations are integrable
and can be mapped to equations with the Painlev\'e property.

INI 1  
12:30 to 14:00  Lunch at Westminster College 