Videos and presentation materials from other INI events are also available.
Event  When  Speaker  Title  Presentation Material 

CATW01 
9th September 2019 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. 

CATW01 
9th September 2019 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. 

CATW01 
9th September 2019 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 $s$. 

CATW01 
9th September 2019 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.


CATW01 
9th September 2019 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). 

CATW01 
10th September 2019 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. 

CATW01 
10th September 2019 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


CATW01 
10th September 2019 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. 

CATW01 
10th September 2019 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.


CATW01 
10th September 2019 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). 

CATW01 
10th September 2019 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.


CATW01 
11th September 2019 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. 

CATW01 
11th September 2019 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. 

CATW01 
11th September 2019 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.


CATW01 
12th September 2019 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. 

CATW01 
12th September 2019 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. 

CATW01 
12th September 2019 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).


CATW01 
12th September 2019 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. 

CATW01 
12th September 2019 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.


CATW01 
12th September 2019 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). 

CATW01 
12th September 2019 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.


CATW01 
13th September 2019 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.


CATW01 
13th September 2019 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.


CATW01 
13th September 2019 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.


CAT 
17th September 2019 14:00 to 15:00 
Andre Weideman  Dynamics of Complex Singularities of Nonlinear PDEs: Analysis and Computation  
CAT 
19th September 2019 16:00 to 17:00 
Marcus Webb  Energy preserving spectral methods on the real line whose analysis strays into the complex plane  
CAT 
24th September 2019 11:00 to 12:00 
Elias Wegert 
Introduction to Nonlinear RiemannHilbert Problems
Nonlinear RiemannHilbert Problems Elias Wegert, TU Bergakademie Freiberg, Germany Though Bernhard Riemann's thesis is commonly known as the source of the celebrated Riemann mapping theorem, Riemann himself considered conformal mapping just as an example to illustrate his ideas about a more general class of nonlinear boundary value problems for analytic functions. The talks aim on making these RiemannHilbert problems more popular, to encourage further research and to find novel applications. In the first part we address the existence and uniqueness of solutions for different problem classes and present two applications: potential flow past a porous object, and a free boundary value problem in electrochemical machining. In the second part, a connection between RiemannHilbert problems and a class of extremal problems is established. Solutions to RiemannHilbert problems are characterized by an extremal principle which generalizes the classical maximum principle and Schwarz' lemma. We briefly sketch an application to the design of dynamical systems. In the end, a class of nonlinear transmission problems is considered. As a special result, we obtain a hyperbolic version of the Riesz decomposition of functions on the unit circle into an analytic and an antianalytic part. 

CAT 
24th September 2019 14:00 to 15:00 
Elias Wegert 
Nonlinear RiemannHilbert Problems (continued)
Nonlinear RiemannHilbert Problems Elias Wegert, TU Bergakademie Freiberg, Germany Though Bernhard Riemann's thesis is commonly known as the source of the celebrated Riemann mapping theorem, Riemann himself considered conformal mapping just as an example to illustrate his ideas about a more general class of nonlinear boundary value problems for analytic functions. The talks aim on making these RiemannHilbert problems more popular, to encourage further research and to find novel applications. In the first part we address the existence and uniqueness of solutions for different problem classes and present two applications: potential flow past a porous object, and a free boundary value problem in electrochemical machining. In the second part, a connection between RiemannHilbert problems and a class of extremal problems is established. Solutions to RiemannHilbert problems are characterized by an extremal principle which generalizes the classical maximum principle and Schwarz' lemma. We briefly sketch an application to the design of dynamical systems. In the end, a class of nonlinear transmission problems is considered. As a special result, we obtain a hyperbolic version of the Riesz decomposition of functions on the unit circle into an analytic and an antianalytic part. 

CAT 
26th September 2019 16:00 to 17:00 
Lesley Ward  Kirk Lecture: Fourier, harmonic analysis, and spaces of homogeneous type  
CAT 
1st October 2019 14:00 to 15:00 
Pavel Lushnikov 
Conformal mapping, Hamiltonian methods and integrability of surface dynamics
A Hamiltonian formulation of the time dependent potential flow of ideal incompressible fluid with a free surface is considered in two dimensional (2D) geometry. It is well known that the dynamics of small to moderate amplitudes of surface perturbations can be reformulated in terms of the canonical Hamiltonian structure for the surface elevation and Dirichlet boundary condition of the velocity potential. Arbitrary large perturbations can be efficiently characterized through a timedependent conformal mapping of a fluid domain into the lower complex halfplane. We reformulate the exact Eulerian dynamics through a noncanonical nonlocal Hamiltonian system for the pair of new conformal variables. The corresponding noncanonical Poisson bracket is nondegenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. We also consider a generalized hydrodynamics for two components of superfluid Helium which has the same noncanonical Hamiltonian structure. In both cases the fluid dynamics is fully characterized by the complex singularities in the upper complex halfplane of the conformal map and the complex velocity. Analytical continuation through the branch cuts generically results in the Riemann surface with infinite number of sheets. An infinite family of solutions with moving poles are found on the Riemann surface. Residues of poles are the constants of motion. These constants commute with each other in the sense of underlying noncanonical Hamiltonian dynamics which provides an argument in support of the conjecture of complete Hamiltonian integrability of surface dynamics. 

CAT 
3rd October 2019 16:00 to 17:00 
Loredana Lanzani 
The joys and pains of multivariable complex analysis: an informal introduction Given the proven success of the complex analysis toolbox in the modeling and solution of problems in Engineering; Physics, etc., we ask: what about multivariable complex analysis? In this talk we provide a broad overview of the basic features of several complex variables with an eye towards building a ``multivariable complex analysis toolbox’’. 

CAT 
8th October 2019 14:00 to 15:00 
Helena Stage 
A heuristic introduction to the applications of WienerHopf factorisation in random processes
Nature abounds with examples of stochastic processes, from scattering in porous media to fluctuations in neuronal signalling. Moving into a more abstract realm, we encounter Lévy processes in probabilistic theory and finance. We will discuss the breadth of examples which can be studied under the lens of stochastic processes, with a focus on random walks. Further, we will illustrate the scope and application of WienerHopf factorisation to these problems by the calculation of a first passage time quantity in a simplified model. 

CAT 
10th October 2019 16:00 to 17:00 
Irina Markina 
Diffeomorphisms of unit circle, shape analysis and some nonlinear PDEs
In the talk, we
explain how univalent functions can be used to analyze plain shapes. In its turn, the univalent functions defined on the unit disc are closely related to the group of oriented preserving diffeomorphisms of the unit circle. A moving plain shape gives rise to a curve on the group of diffeomorphisms. The requirement to describe a shape modulo its rotation and/or scaling leads to a curve subordinated to some constraints. A geodesic curve of the motion of a shape is a solution to some nonlinear partial differential equation. The choice of metric leads to different PDEs, that are generalizations of equations originated in fluid dynamics, such us inviscid Burgers' equation, CamassaHolm, HunterSaxton, and KdV. 

CAT 
17th October 2019 16:00 to 17:00 
Matthew Turner 
Timedependent conformal mapping techniques applied to fluid sloshing problems In this presentation we demonstrate how timedependent conformal mappings can be used to construct a fast and effective numerical scheme for examining twodimensional, inviscid, irrotational fluid sloshing in both fixed and moving vessels. In particular we examine how horizontal sidewall baffles can be incorporated into the system creating a multiplyconnected fluid domain. 

CAT 
24th October 2019 10:30 to 12:00 
Saleh Tanveer  Freeboundary problems, singularities and exponential asymptotics  
CAT 
24th October 2019 14:00 to 15:30 
Alexander Its 
The RiemannHilbert method. Toeplitz determinants as a case study
The RiemannHilbert method is one of the primary analytic tools of modern theory of integrable systems. The origin of the method goes back to Hilbert's 21st prob lem and classical WienerHopf method. In its current form, the RiemannHilbert approach exploits ideas which goes beyond the usual WienerHopf scheme, and they have their roots in the inverse scattering method of soliton theory and in the theory of isomonodromy deformations. The main \beneciary" of this, latest ver sion of the RiemannHilbert method, is the global asymptotic analysis of nonlinear systems. Indeed, many longstanding asymptotic problems in the diverse areas of pure and applied math have been solved with the help of the RiemannHilbert technique. One of the recent applications of the RiemannHilbert method is in the theory of Toeplitz determinants. Starting with Onsager's celebrated solution of the two dimensional Ising model in the 1940's, Toeplitz determinants have been playing an increasingly important role in the analytic apparatus of modern mathematical physics; specically, in the theory of exactly solvable statistical mechanics and quantum eld models. In these two lectures, the essence of the RiemannHilbert method will be pre sented taking the theory of Topelitz determinants as a case study. The focus will be on the use of the method to obtain the Painleve type description of the tran sition asymptotics of Toeplitz determinants. The RIemannHilbert view on the Painleve functions will be also explained. 

CAT 
24th October 2019 16:00 to 17:00 
Irina Mitrea 
Harmonic Analysis on Uniformly Rectifiable Sets and Applications to Complex Analysis of a Single and Several Variables
In this talk I will discuss recent developments at the interface
between Harmonic Analysis and Geometric Measure Theory (Calder\'onZygmund
theory on Uniformly Rectifiable sets,
a sharp divergence theorem with nontangential boundary traces, Fatoutype
theorems, etc.) and present their impact in problems arising in Complex
Analysis, of a single and several variables, which make systematic use of
singular
integral operators.


CAT 
25th October 2019 10:30 to 12:00 
Saleh Tanveer  Freeboundary problems, singularities and exponential asymptotics  
CAT 
25th October 2019 14:00 to 15:30 
Alexander Its 
The RiemannHilbert method. Toeplitz determinants as a case study
The RiemannHilbert method is one of the primary analytic tools of modern theory of integrable systems. The origin of the method goes back to Hilbert's 21st prob lem and classical WienerHopf method. In its current form, the RiemannHilbert approach exploits ideas which goes beyond the usual WienerHopf scheme, and they have their roots in the inverse scattering method of soliton theory and in the theory of isomonodromy deformations. The main \beneciary" of this, latest ver sion of the RiemannHilbert method, is the global asymptotic analysis of nonlinear systems. Indeed, many longstanding asymptotic problems in the diverse areas of pure and applied math have been solved with the help of the RiemannHilbert technique. One of the recent applications of the RiemannHilbert method is in the theory of Toeplitz determinants. Starting with Onsager's celebrated solution of the two dimensional Ising model in the 1940's, Toeplitz determinants have been playing an increasingly important role in the analytic apparatus of modern mathematical physics; specically, in the theory of exactly solvable statistical mechanics and quantum eld models. In these two lectures, the essence of the RiemannHilbert method will be pre sented taking the theory of Topelitz determinants as a case study. The focus will be on the use of the method to obtain the Painleve type description of the tran sition asymptotics of Toeplitz determinants. The RIemannHilbert view on the Painleve functions will be also explained. 

CATW02 
28th October 2019 10:00 to 11:00 
Martine Ben Amar 
Complex analysis and nonlinear elasticity: the Biot instability revisited.
The Biot instability is probably the most useful tool to explain many features in morphogenesis of soft tissues. It explains pretty well the circumvolutions of the brain, the villi of our intestine or our fingerprints.
Nevertheless, the theory remains technically difficult for two main reasons: the constraint of incompressibility but also the geometry of finite samples.
For the simplest case, that is the semiinfinite NeoHookean sample under growth or compression, I will show that complex analysis offers a simple way to treat the buckling instability
at the Biot threshold,but also above and below the threshold.


CATW02 
28th October 2019 11:30 to 12:30 
John King 
Complexplane analysis of blow up in reaction diffusion The benefits and difficulties of extending the analysis of blow up into the complex plane will be illustrated. 

CATW02 
28th October 2019 13:30 to 14:00 
Clarissa Schoenecker 
Flows close to patterned, slippery surfaces
Micro or nanostructured surfaces can provide a significant slip to a fluid flowing over the surface, making them attractive for the development of functional coatings. This slip is due to a second fluid being entrapped in the indentations of the structured surface, like air for superhydrophobic surfaces or oil for socalled lubricantinfused surfaces (SLIPS). This talk addresses the flow phenomena close to such surfaces. The nature of the structured surface leads to a mixedboundary value problem for the flow field, which, for the considered situation, obeys the biharmonic or Laplace equation. Using complex variables techniques, a solution to this problem can be found, such that simple, explicit expressions for the flow field as well as the effective slip length of the surface can be found. They can for example be employed as a guideline to design efficient surface coatings for drag reduction. This is work together with Steffen Hardt (TU Darmstadt).


CATW02 
28th October 2019 14:00 to 14:30 
Oleg Lisovyy 
Painlevé functions, accessory parameters and conformal blocks
We will consider three a priori different types of special functions : (i) tau functions of Painlevé equations (ii) accessory parameters in linear differential equations of Heun’s type and (iii) conformal blocks of the Virasoro algebra. I wil discuss surprising (and for the most part conjectural) relations between these functions coming from the 2D conformal field theory.


CATW02 
28th October 2019 14:30 to 15:30 
Ken McLaughlin 
Asymptotic analysis of RiemannHilbert problems and applications
It is hoped that 1/2 of this presentation will be introductory, explaining with examples how RiemannHilbert problems characterize solutions of some interesting questions in mathematical physics, and some of the complex variables techniques that are exploited. The presentation will end with an explanation of results with Manuela Girotti (John Abbot College, Montreal), Tamara Grava (Bristol and SISSA), and Robert Jenkins (Univ. of Central Florida) concerning the asymptotic behavior of an infinite collection of solitons under the Korteweg de Vries equation.


CATW02 
28th October 2019 16:00 to 17:00 
Dave Smith 
Linear evolution equations with dynamic boundary conditions
The classical half line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this talk, we study the problem in which the static Robin condition $bq(0,t)+q_x(0,t)=0$ is replaced with a dynamic Robin condition; $b=b(t)$ is allowed to vary in time. We present a solution representation, and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation, and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half line, with arbitrary linear dynamic boundary conditions.


CATW02 
29th October 2019 09:00 to 10:00 
Scott McCue 
Applying conformal mapping and exponential asymptotics to study translating bubbles in a HeleShaw cell In a traditional HeleShaw configuration, the governing equation for the pressure is Laplace's equation; thus, mathematical models for HeleShaw flows are amenable to complex analysis. We consider here one such problem, where a bubble is moving steadily in a HeleShaw cell. This is like the classical TaylorSaffman bubble, except we suppose the domain extends out infinitely far in all directions. By applying a conformal mapping, we produce numerical evidence to suggest that solutions to this problem behave in an analogous way to wellstudied finger and bubble problems in a HeleShaw channel. However, the selection of the ratio of bubble speeds to background velocity for our problem appears to follow a very different surface tension scaling to the channel cases. We apply techniques in exponential asymptotics to solve the selection problem analytically, confirming the numerical results, including the predicted surface tension scaling laws. Further, our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane. These results are likely to provide insight into other wellknown selection problems in HeleShaw flows. 

CATW02 
29th October 2019 10:00 to 11:00 
Nick Moore 
How Focused Flexibility Maximizes the Thrust Production of Flapping Wings
Birds, insects, and fish all exploit the fact that flexible wings or fins generally perform better. It is not clear, though, how to best distribute flexibility: Should a wing be uniformly flexible, or should certain sections be more rigid than others? I will discuss this question by using a smallamplitude model combined with an efficient Chebyshev PDE solver that exploits the 2D nature of the problem through conformal mapping techniques. Numerical optimization shows that concentrating flexibility near the leading edge of the wing maximizes thrust production.


CATW02 
29th October 2019 11:30 to 12:30 
Andrei MartinezFinkelshtein 
Spectral curves, variational problems, and the hermitian matrix model with external source
We show that to any cubic equation from a special class (`a "spectral curve") it corresponds a unique vectorvalued measure with three components on the complex plane, characterized as a solution of a variational problem stated in terms of their logarithmic energy. We describe all possible geometries of the supports of these measures: the third component, if nontrivial, lives on a contour on the plane and separates the supports of the other two measures, both on the real line. This general result is applied to the hermitian random matrix model with external source and general polynomial potential, when the source has two distinct eigenvalues but is otherwise arbitrary. We prove that under some additional assumptions any limiting zero distribution for the average characteristic polynomial can be written in terms of a solution of a spectral curve. Thus, any such limiting measure admits the above mentioned variational description. As a consequence of our analysis we obtain that the density of this limiting measure can have only a handful of local behaviors: Sine, Airy and their higher order type behavior, Pearcey or yet the fifth power of the cubic (but no higher order cubics can appear). This is a joint work with Guilherme Silva (U. Michigan, Ann Arbor). We also compare our findings with the most general results available in the literature, showing that once an additional symmetry is imposed, our vector critical measure contains enough information to recover the solutions to the constrained equilibrium problem that was known to describe the limiting eigenvalue distribution in this symmetric situation. 

CATW02 
29th October 2019 13:30 to 14:30 
Maxim Yattselev 
On multiple orthogonal polynomials
I will discuss certain algebraic and analytic
properties of polynomials (multiple) orthogonal with respect to a pair of
measures on the real line.


CATW02 
29th October 2019 14:30 to 15:30 
Tom Claeys 
The lower tail of the KPZ equation via a RiemannHilbert approach
Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the KardarParisiZhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distribution in models of positivetemperature free fermions. I will explain how logarithmic derivatives of the Fredholm determinants can be expressed in terms of a $2\times 2$ RiemannHilbert problem, and how we can use this to derive asymptotics for the Fredholm determinants. As an application of our result, we derive precise lower tail asymptotics for the solution of the KPZ equation with narrow wedge initial data which refine recent results by Corwin and Ghosal.


CATW02 
29th October 2019 16:00 to 17:00 
Loredana Lanzani 
Cauchytype integrals in multivariable complex analysis This is joint work with Elias M. Stein (Princeton University). 

CATW02 
31st October 2019 09:00 to 10:00 
Peter Clarkson 
Rational solutions of three integrable equations and applications to rogue waves In this talk I shall discuss rational solutions of the Boussinesq equation, the focusing nonlinear Schr\"odinger (NLS) equation and the KadomtsevPetviashvili I (KPI) equation, which are all soliton equations solvable by the inverse scattering. 

CATW02 
31st October 2019 10:00 to 11:00 
Igor Krasovsky 
Hausdorff dimension of the spectrum of the almost Mathieu operator
We
will discuss the wellknown quasiperiodic operator: the almost Mathieu
operator
in the critical case. We give a new and elementary proof (the first proof was
completed in 2006 by Avila and Krikorian by a different method) of the fact
that its spectrum is a zero measure Cantor set. We furthermore prove a
conjecture going back to the work of David Thouless in 1980s, that the
Hausdorff dimension of the spectrum is not larger than 1/2. This is a joint
work with Svetlana Jitomirskaya.


CATW02 
31st October 2019 11:30 to 12:30 
Michael Siegel 
Complex variable techniques applied to two problems in Stokes flow: rotation of a superhydrophobic cylinder, and evolution of multiple drops with surfactant
We present two problems in which complex variable techniques provide an essential component of the analysis. First, a model of a superhydrophobic cylinder rotating in a viscous liquid is considered. The boundary of the cylinder is assumed to contain alternating noslip and noshear surfaces oriented transverse to the flow. The main interest is in computing the hydrodynamic torque on the cylinder. An explicit solution to the flow problem is obtained by combining complex variable techniques with asymptotic analysis. This work is joint with Ehud Yariv (Technion). Second, we present a new boundary integral method for computing the evolution of multiple surfactant covered drops in 2D Stokes flow. The method maintains high accuracy when drops are in very close proximity, which is a challenging situation for numerical computation. Complex variable techniques are critical in both the design and validation of the method. This is joint work with Sara Palsson and AnnaKarin Tornberg (KTH Stockholm). 

CATW02 
31st October 2019 13:30 to 14:30 
Anna Zemlyanova 
Contact problems at nanoscale
In this talk, the surface elasticity in the form proposed by Steigmann and Ogden is applied to study plane problems of frictionless or adhesive contact of a rigid stamp with an elastic upper semiplane. The results of the present work generalize the results for contact problems with GurtinMurdoch elasticity by including additional dependency on the curvature of the surface. The mechanical problem is reduced to a system of singular integrodifferential equations which is regularized using Fourier transform method. The sizedependency of the solutions of the problem is highlighted. It is observed that the curvaturedependence of the surface energy is increasingly important at small scales. The numerical results are presented for different values of the mechanical parameters.


CATW02 
31st October 2019 14:30 to 15:30 
Robb McDonald 
Growth of thin fingers in Laplacian and Poisson fields (i) The Laplacian growth of thin twodimensional protrusions in the form of either straight needles or curved fingers satisfying Loewner's equation is studied using the SchwarzChristoffel (SC) map. Particular use is made of Driscoll's numerical procedure, the SC Toolbox, for computing the SC map from a halfplane to a slit halfplane, where the slits represent the needles or fingers. Since the SC map applies only to polygonal regions, in the Loewner case, the growth of curved fingers is approximated by an increasing number of short straight line segments. The growth rate of the fingers is given by a fixed power of the harmonic measure at the finger or needle tips and so includes the possibility of ‘screening’ as they interact with themselves and with boundaries. The method is illustrated by examples of needle and finger growth in halfplane and channel geometries. Bifurcating fingers are also studied and application to branching stream networks discussed. 

CATW02 
31st October 2019 16:00 to 17:00 
Kevin O'Neil 
Stationary vortex sheets and limits of polynomials Stationary distributions of vorticity in a twodimensional fluid are considered where the vorticity support consists of points (point vortices) and curves (vortex sheets.) The flow in many cases is determined by a simple rational function. By comparing the vorticity distributions to point vortex configurations associated with polynomials, these distributions can be related other questions such as nonuniqueness of HeineStieltjes polynomials and to the asymptotics of some nonclassical special cases of orthogonal polynomials. 

CATW02 
1st November 2019 09:00 to 10:00 
Craig Tracy  Blocks in the asymmetric simple exclusion process  
CATW02 
1st November 2019 10:00 to 11:00 
Vladimir Mityushev 
Cluster method in the theory of fibrous elastic composites Consider a 2D multiphase random composite with different circular inclusions. A finite number $n$ of inclusions on the infinite plane forms a cluster. The corresponding boundary value problem for Muskhelishvili's potentials is reduced to a system of functional equations. Solution to the functional equations can be obtained by a method of sucessive approximations or by the Taylor expansion of the unknown analytic functions. Next, the local stressstrain fields are calculated and the averaged elastic constants are obtained in symbolic form. Extensions of Maxwell's approach and other various selfconsisting methods are discussed. An uncertainty when the number of elements $n$ in a cluster tends to infinity is analyzed by means of the conditionally convergent series. Basing on the fields around a finite cluster without clusters interactions one can deduce formulae for the effective constants only for dilute clusters. The Eisenstein summation yields new analytical formulae for the effective constants for random 2D composites with high concentration of inclusions. 

CATW02 
1st November 2019 11:30 to 12:30 
Takashi Sakajo 
Vortex dynamics on the surface of a torus
As theoretical models of incompressible flows arising in engineering and geophysical problems, vortex dynamics is sometimes considered on surfaces that have various geometric features such as multiply connected domains and spherical surfaces. The models are derived from the streamlinevorticity formulation of the Euler equations. In order to solve the model equations, complex analysis and its computational techniques are effectively utilized. In the present talk, we consider vortex dynamics on the surface of a torus. Although the flows on the surface of a torus is no longer a physical relevance to real fluid flow phenomena, it is theoretically interesting to observe whether the geometric nature of the torus, i.e., a compact, orientable 2D Riemannian manifold with nonconstant curvature and one handle, yields different vortex dynamics that are not observed so far. The vortex model is not only an intrinsic theoretical extension in the field of classical fluid mechanics, but it would also be applicable to modern physics such as quantum mechanics and flows of superfluid films. Based on the model of point vortices, where the vorticity distribution is given by discrete delta measures, we investigate equilibrium states of point vortices, called vortex crystals, moving in the longitudinal direction without changing their relative configuration. Moreover, we derive an analytic solution of a modified Liouville equation on the toroidal surface, where the vorticity distribution is given by an exponential of the streamfunction. The solution gives rise to a vortex crystal with quantized circulations embedded in a continuous vorticity distribution in the plane, which corresponds to a model of shear flows in the plane known as Stuart vortex. A part of the results presented in this talk is based on the joint works with Mr. Yuuki Shimizu, Kyoto University.


CATW02 
1st November 2019 13:30 to 14:30 
Yuri Antipov 
Diffraction by wedges: higher order boundary conditions, integral transforms, vector RiemannHilbert problems, and Riemann surfaces
Acoustic and electromagnetic diffraction by a wedge is modeled by one and two Helmholtz equations coupled by boundary conditions. When the wedge walls are membranes or elastic plates, the impedance boundary conditions have derivatives of the third or fifth order, respectively. A new method of integral transforms for rightangled wedges is proposed. It is based on application of two Laplace transforms. The main feature of the method is that the second integral transform parameter is a specific root of the characteristic polynomial of the ordinary differential operator resulting from the transformed PDE by the first Laplace transform. For convex domains (concave obstacles), the problems reduce to scalar and order2 vector RiemannHilbert problems. When the wedge is concave (a convex obstacle), the acoustic problem is transformed into an order3 RiemannHilbert problem. The order2 and 3 vector RiemannHilbert problems are solved by recasting them as scalar RiemannHilbert problems on Riemann surfaces. Exact solutions of the problems are determined. Existence and uniqueness issues are discussed.


CATW02 
1st November 2019 14:30 to 15:30 
Gennady Mishuris 
Analyses of dynamic fault propagation in discrete structures
We discuss a method proposed by L. Slepyan in 1981 which allows solving various problems related to wave and fracture propagation in discrete structures. It has already shown its effectiveness to tackle a wide range of the problems and structures. One can mention: full/bridge crack propagation, phase transition; various lattice (rectangular, triangular) and chain structures; various links (springs / beams), etc. Such problems are eventually reduced to scalar WienerHopf problems. This allows to solve them and extract valuable information about the processes. We discuss both advantages and limitations of the method and highlight its importance for metamaterials. We conclude by revisiting a few "simplest" problems demonstrating recent advances in the area. 

CATW02 
1st November 2019 16:00 to 17:00 
Xie Xuming 
Existence results in interfacial flows with kinetic undercooling regularization in a timedependent gap HeleShaw cell HeleShaw cells where the top plate is lifted uniformly at a prescribed speed and the bottom plate is fixed have been used to study interface related problems. Our talk focuses on an interface flow with kinetic undercooling regularization in a radial HeleShaw cell with a time dependent gap. We obtain the local existence of analytic solution of the moving boundary problem when the initial data is analytic. The methodology is to use complex analysis and reduce the free boundary problem to a RiemannHilbert problem and an abstract CauchyKovalevsky evolution problem. A similar free boundary problem in multidimensional spaces is also studied and the local existence of classical solutions can be obtained. 

CAT 
7th November 2019 16:00 to 17:00 
Nick Moore 
Experiments ad theory for anomalous waves induced by abrupt depth changes
I will discuss both laboratory experiments and a newly developed theory
for randomized surface waves propagating over variable bathymetry. The
experiments show that an abrupt depth change can qualitatively alter wave
statistics, transforming an initially Gaussian wave field into a highly skewed
one. In our experiments, the probability of a rogue wave can increase by a
factor of 50 compared to what would be expected from normal statistics. I will
discuss a theoretical framework based on dynamical and statistical analysis of
the truncated KdV equations. This theory accurately captures many key features
of the experiments, such as the skewed outgoing wave distributions and the
associated excitation of higher frequencies in the spectrum. 

CAT 
12th November 2019 14:00 to 15:00 
Robert Corless 
An Older Special Function meets a (Slightly) Newer One
Euler
invented the Gamma function in 1729, and it remains one of the moststudied
special functions; see in particular Philip J. Davis'
Chauvenet
prizewinning article "Leonhard Euler's Integral", 1959. In
2016, Jon Borwein and I started a survey of articles on Gamma in the American
Mathematical Monthly (including that beautiful paper by Davis); Jon died before
our survey was finished, but I finished it and it was published in 2018:
"Gamma and Factorial in the Monthly". In that survey, we uncovered a
surprising gap in the nearly three hundred years of literature subsequent to
Euler's invention: almost nobody had studied the functional inverse of the
Gamma function. More, we uncovered Stirling's original asymptotic series
(the asymptotic series that "everyone knows" as Stirling's is, in
fact, due to de Moivre), and used it to find a remarkably accurate approximation
to the principal branch of the functional inverse of Gamma using what is now
known as the Lambert W function. This newer function was also invented by
Euler, in 1783, using a series due to Lambert in 1758; since Euler did not need
yet another function or equation named after him, we chose in the mid 1990's to
name it after Lambert. Facts about W may be found at http://www.orcca.on.ca/LambertW and
physical copies of that poster are being couriered here; not, unfortunately in
time for the talk, but you'll be able to get your very own copy (lucky you!)
probably by the end of the week.
My talk
will survey some of the Monthly articles on Gamma, and introduce some of the
facts about W that I find interesting. I will have more material than
will fit in an hour, so exactly which topics get covered will depend at least
in part on the interests of the audience.
The
Monthly paper can be found at
https://www.tandfonline.com/doi/full/10.1080/00029890.2018.1420983
The
Wikipedia articles on Gamma and on Lambert W are substantive.


CAT 
14th November 2019 16:00 to 17:00 
Mark Ablowitz 
Rothschild Lecture: Extraordinary waves and math: from beaches to photonics Waves are fascinating. They are of keen interest across the realm of science and mathematics and they often elicit excitement in the general public. There is a class of extraordinary localized waves, called solitary waves, that were first observed 185 years ago. More recently certain solitary waves, often called solitons, have been found to possess numerous special properties. This lecture will discuss the history of these waves, and will indicate why mathematics played a crucial role in both historical and modern developments. Some applications will be described including as time permits: water waves, rogue waves, photonics and cellular automata. The discussion will be general; it will `leave almost all equations behind'. 

CAT 
15th November 2019 11:00 to 12:00 
Thanasis Fokas 
The Unified Transform, Medical Imaging, Asymptotics of the Riemann Zeta Function: Part I
Employing techniques of complex analysis, three different problems will be discussed: (i) Initialboundary value problems via the unified transform (also known as the Fokas method,www.wikipedia.org/wiki/Fokas_method)[1]. (ii) The evaluation of the large tasymptotics to all orders of the Riemann zeta function[2], and the introduction of a new approach to the Lindelöf Hypothesis[3]. (iii) A novel analytical algorithm for the medical technique of SPECT and its numerical implementation [4]. [1] J. Lenells and A. S. Fokas. The Nonlinear Schrödinger Equation with tPeriodic Data: I. Exact Results, Proc. R. Soc. A 471, 20140925 (2015). J. Lenells and A. S. Fokas, The Nonlinear Schrödinger Equation with tPeriodic Data: II. Perturbative Results, Proc. R. Soc. A 471, 20140926 (2015). [2] A.S. Fokas and J. Lenells, On the Asymptotics to All Orders of the Riemann Zeta Function and of a TwoParameter Generalization of the Riemann Zeta Function, Mem. Amer. Math. Soc. (to appear). [3] A.S. Fokas, A Novel Approach to the Lindelof Hypothesis, Transactions of Mathematics and its Applications, 3(1), tnz006 (2019). [4]N.E. Protonotarios, A.S. Fokas, K. Kostarelos and G.A. Kastis, The Attenuated Spline Reconstruction Technique for Single Photon Emission Computed Tomography, J. R. Soc. Interface 15, 20180509 (2018). 

CAT 
15th November 2019 14:00 to 15:00 
Thanasis Fokas 
The Unified Transform, Medical Imaging, Asymptotics of the Riemann Zeta Function: Part II
Employing techniques of complex analysis, three different problems will be discussed: (i) Initialboundary value problems via the unified transform (also known as the Fokas method,www.wikipedia.org/wiki/Fokas_method)[1]. (ii) The evaluation of the large tasymptotics to all orders of the Riemann zeta function[2], and the introduction of a new approach to the Lindelöf Hypothesis[3]. (iii) A novel analytical algorithm for the medical technique of SPECT and its numerical implementation [4]. [1] J. Lenells and A. S. Fokas. The Nonlinear Schrödinger Equation with tPeriodic Data: I. Exact Results, Proc. R. Soc. A 471, 20140925 (2015). J. Lenells and A. S. Fokas, The Nonlinear Schrödinger Equation with tPeriodic Data: II. Perturbative Results, Proc. R. Soc. A 471, 20140926 (2015). [2] A.S. Fokas and J. Lenells, On the Asymptotics to All Orders of the Riemann Zeta Function and of a TwoParameter Generalization of the Riemann Zeta Function, Mem. Amer. Math. Soc. (to appear). [3] A.S. Fokas, A Novel Approach to the Lindelof Hypothesis, Transactions of Mathematics and its Applications, 3(1), tnz006 (2019). [4]N.E. Protonotarios, A.S. Fokas, K. Kostarelos and G.A. Kastis, The Attenuated Spline Reconstruction Technique for Single Photon Emission Computed Tomography, J. R. Soc. Interface 15, 20180509 (2018). 

CAT 
19th November 2019 04:00 to 05:00 
Jonathan Chapman  Some problems in exponential asymptotics  
CAT 
21st November 2019 16:00 to 17:00 
Peter Clarkson  Symmetric Orthogonal Polynomials  
CAT 
26th November 2019 14:00 to 15:00 
Scott McCue 
Some old and new moving boundary problems for HeleShaw flow
When a less viscous fluid is injected into a more viscous fluid in a
HeleShaw, the interface between the fluids is unstable. This is the
SaffmanTaylor instability which can give rise to a striking pattern formation
at the interface. The usual model for this type of flow is a
twodimensional moving boundary problem which is amenable to complex variable
methods, especially in the special case in which surface tension is ignored.
For nonstandard geometries progress can be made via numerical simulation,
using a level set formulation for example. Some recent experimental,
analytical and numerical studies of nonstandard moving boundary problems in
HeleShaw flow will be discussed. This is joint work with Liam Morrow,
Tim Moroney and Michael Dallaston.


CAT 
27th November 2019 11:00 to 12:00 
Yuri Antipov 
RiemannHilbert problems of the theory of automorphic functions and inverse problems of elasticity and cavitating flow for multiply connected domains
The general theory of the
RiemannHilbert problem for piecewise holomorphic automorphic functions
generated by the Schottky symmetry groups is discussed. The theory is
illustrated by two inverse problems for multiply connected domains. The first
one concerns the determination of the profiles on n inclusions in an elastic
plane subjected to shear loading at infinity when the stress field in the
inclusions is uniform. The second problem is a model problem of cavitating flow
past n hydrofoils. Both problems are solved by the method of conformal
mappings. The maps from nconnected circular domain into the physical domain
are reconstructed by solving two RiemannHilbert problems of the theory of
piecewise holomorphic automorphic functions. 

CAT 
3rd December 2019 14:00 to 15:00 
Sonia Mogilevskaya 
Lost in Translation: Crack Problems in Different Languages
The feeling of “lost in translation” is familiar to everyone
who stumbles on a relevant literature source written by someone with a
different academic background. It may take a significant effort to “translate”
the source and interpret its contents into a familiar “language.” This may
happen in various research areas and analysis of crack problems is one of such
examples. The methods of analytical or
computational crack modelling vary with the academic background of the
researcher and with specific applications. In addition, as typically happens in
scientific research, the developments in the relevant areas take place
simultaneously in different countries, and the results are literally described
and published in different languages. Some of those publications are not even
translated.
The talk examines major techniques for modelling elastostatic
crack problems. The foundations of these techniques and fundamental papers that
introduced, developed, and applied them are reviewed. The goal is to provide
“translation” between different academic languages that describe the same
problem.


CAT 
5th December 2019 16:00 to 17:00 
Mark Blyth 
Towards a model of a deformable aerofoil
Aerofoils that include flexible components are designed to enhance aerodynamic facility and to promote fuel efficiency under a range of different flight conditions. Some small, unmanned aerial vehicles (UAVs) use entirely collapsible, elastic wings with a view to ready portability. In this talk we work toward a model of a wholly deformable aerofoil, considered as a thinwalled elastic cell placed into a uniform stream. We use a conformal mapping approach to determine the cell shape and the ambient flow simultaneously. Our initial analysis of a light cell in an oncoming flow can be viewed as a generalisation of Flaherty et al.'s 1972 work on the buckling of an elastic cell under a constant transmural pressure difference. Introducing cell mass and circulation/lift, we study equilibrium shapes at different flow speeds and for different transmural pressure jumps. A fixedangle corner at the trailing edge is introduced by way of a KarmanTrefftz conformal mapping, and an internal strut is included, to more accurately mimic the shape and aerodynamic properties of a traditional, rigid aerofoil. DNS simulations are carried out to assess the performance of the equilibrium aerofoil shapes in a real flow.


CATW03 
9th December 2019 10:00 to 11:00 
Nick Trefethen 
Masterclass: contour integrals Part 1 of a threehour series on Complex Computing in MATLAB and Chebfun 

CATW03 
9th December 2019 11:30 to 12:30 
Daan Huybrechs 
PathFinder: a toolbox for oscillatory integrals by deforming into the complex plane We present PathFinder, a Matlab toolbox developed by Andrew Gibbs for the computation of oscillatory integrals. The methods are based on the numerical evaluation of line integrals in the complex plane, which mostly follow paths of steepest descent. The toolbox aims for automation and robustness: the path deformation is found fully automatically, but the paths correspond to those of steepest descent only if doing so has guarantees of correctness. We face a tradeoff between achieving asymptotic orders of accuracy, i.e. increasing accuracy with increasing frequency much like asymptotic expansions, and reliable accuracy. In the toolbox we aim for the former, but give precedence to the latter. We conclude with an example of scattering integrals with phase functions that may have clusters of coalescing saddle points. 

CATW03 
9th December 2019 13:30 to 14:00 
Alex Townsend 
Chebyshev to Zolotarev, Faber to Ganelius, and EIM to AAA
In 1854, Chebyshev derived the Chebyshev polynomials via a minimax polynomial problem. About 20 years later, Zolotarev (a student in one of Chebyshev's courses) generalized the minimax problem to one involving rational functions. These minimax problems are now used to understand the convergence behavior of Krylov methods, the decay rate of singular values of structured matrices, and the development of fast PDE solvers. In this talk, we will survey the computational complex analysis techniques that can be used to solve Chebyshev's and Zolotarev's minimax problems and try to highlight the ongoing connections between polynomials and rationals.


CATW03 
9th December 2019 14:00 to 14:30 
Nick Hale 
Numerical Aspects of Quadratic Padé Approximation
A classical (linear) Padé approximant is a rational approximation, F(x) = p(x)/q(x), of a given function, f(x), chosen so the Taylor series of F(x) matches that of f(x) to as many terms as possible. If f(x) is meromorphic, then F(x) often provides a good approximation of f(x) in the complex plane beyond the radius of convergence of the original Taylor series. A generalisation of this idea is quadratic Padé approximation, where now polynomials p(x), q(x), and r(x) are chosen so that p(x) + q(x)f(x) + r(x)f^2(x) = O(x^{max}). The approximant, F(x), can then be found by solving p(x) + q(x)F(x) + r(x)F^2(x) = 0 by, for example, the quadratic formula. Since F(x) now contains branch cuts, it typically provides better approximations than linear Padé approximants when f(x) is multisheeted, and may be used to estimate branch point locations as well as poles and roots of f(x).
In this talk we focus not on approximation properties of Padé approximants, but rather on numerical aspects of their computation. In the linear case things are wellunderstood. For example, it is wellknown that the ill conditioning in the linear system satisfied by p(x) and q(x) means that these are computed with poor relative error, but that in practice, F(x) itself still has good relative accuracy. Luke (1980) formalises this for linear Padé approximants, and we show how this analysis extends to the quadratic case. We discuss a few different algorithms for computing a quadratic Padé approximation, explore some of the problems which arise in the evaluation of the approximant, and demonstrate some example applications.


CATW03 
9th December 2019 14:30 to 15:00 
Andrew Gibbs 
Numerical steepest descent for singular and oscillatory integrals CoAuthors: Daan Huybrechs, David Hewett 

CATW03 
9th December 2019 15:00 to 15:30 
Matthew Colbrook 
The Foundations of InfiniteDimensional Spectral Computations
Spectral computations in infinite dimensions are ubiquitous in the sciences and the problem of computing spectra is one of the most studied areas of computational mathematics over the last halfcentury. However, such computations are infamously difficult, since standard approaches do not, in general, produce correct solutions (the most famous problem in the selfadjoint case is spectral pollution in gaps of the essential spectrum). The goal of this talk is to introduce classes of resolvent based algorithms that compute spectral properties of operators on separable Hilbert spaces. As well as solving computational problems for the first time, these algorithms are proven to be optimal, and the computational problems themselves can be classed in a hierarchy (the SCI hierarchy) with ramifications beyond spectral theory. For concreteness, I shall focus on two problems for a very general class of operators on $l^2(\mathbb{N})$, where algorithms access the matrix elements of the operator: 1) Computing spectra of closed operators in the AttouchWets topology (local uniform convergence of closed sets). This algorithm uses estimates of the norm of the resolvent operator and a local minimisation scheme. As well as solving the longstanding computational spectral problem, this algorithm computes spectra with error control. It can also be extended to partial differential operators with coefficients of locally bounded total variation with algorithms point sampling the coefficients. 2) Computing (projectionvalued) spectral measures of selfadjoint operators as given by the spectral theorem. This algorithm uses computation of the full resolvent operator (with asymptotic error control) to compute convolutions of rational kernels with the measure before taking a limit. I shall discuss local convergence properties and extensions to computing spectral decompositions (pure point, absolutely continuous and singular continuous parts). Finally, these algorithms are embarrassingly parallelisable. Numerical examples will be given, demonstrating efficiency, and tackling difficult problems taken from mathematics and other fields such as chemistry and physics. 

CATW03 
9th December 2019 16:00 to 16:30 
Anastasia Kisil 
Effective ways of solving some PDEs with complicated Boundary Conditions on Unbounded domains This talk will be split into two parts. 

CATW03 
9th December 2019 16:30 to 17:00 
Elena Luca 
Numerical methods for mixed boundary value problems in diffraction and homogenization theory
In this talk, we present fast and accurate numerical methods for the solution of mixed boundary value problems and of the associated matrix Wiener–Hopf problems. The Wiener–Hopf problems are formulated as Riemann–Hilbert problems on the real line, and the numerical approach for such problems of Trogdon & Olver (2015) is employed. It is shown that the known farfield behaviour of the solutions can be exploited to construct tailormade numerical schemes providing accurate results. A number of scalar and matrix Wiener–Hopf problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semiinfinite plane, as well as problems arising in homogenization theory, are solved using the new approach.


CATW03 
10th December 2019 09:00 to 09:30 
Maria Christina van der Weele 
Integrable Systems in Multidimensions
One of the main current topics in the field of integrable systems concerns the existence of nonlinear integrable evolution equations in more than two spatial dimensions. The fact that such equations exist has been proven by A.S. Fokas [1], who derived equations of this type in four spatial dimensions, which however had the disadvantage of containing two time dimensions. The associated initial value problem for such equations, where the dependent variables are specified for all space variables at t1 = t2 = 0, can be solved by means of a nonlocal dbar problem. The next step in this program is to formulate and solve nonlinear integrable systems in 3+1 dimensions (i.e., with three space variables and a single time variable) in agreement with physical reality. The method we employ is to first construct a system in 4+2 dimensions, with the aim to reduce this then to 3+1 dimensions. In this talk we focus on the DaveyStewartson system [2] and the 3wave interaction equations [3]. Both these integrable systems have their origins in fluid dynamics where they describe the evolution and interaction, respectively, of wave packets on e.g. a water surface. We start from these equations in their usual form in 2+1 dimensions (two space variables x, y and one time variable t) and we bring them to 4+2 dimensions by complexifying each of these variables. We solve the initial value problem of these equations in 4+2 dimensions. Subsequently, in the linear limit we reduce this analysis to 3+1 dimensions to comply with the natural world. Finally, we discuss the construction of the 3+1 reduction of the full nonlinear problem, which is currently under investigation. This is joint work together with my PhD supervisor Prof. A.S. Fokas. References [1] A.S. Fokas, Integrable Nonlinear Evolution Partial Differential Equations in 4+2 and 3+1 Dimensions, Phys. Rev. Lett. 96 (2006), 190201. [2] A.S. Fokas and M.C. van der Weele, Complexification and integrability in multidimensions, J. Math. Phys. 59 (2018), 091413. [3] M.C. van der Weele and A.S. Fokas, Solving the Initial Value Problem for the 3Wave Interaction Equations in Multidimensions (to be submitted, 2019). 

CATW03 
10th December 2019 09:30 to 10:00 
Percy Deift 
Universality in numerical analysis. Cyber algorithms
It turns out that
for a wide variety of numerical algorithms with random
data, the stopping times for the algorithms to achieve a given
accuracy, have
universal fluctuations independent of the ensemble for the random data. The speaker
will discuss
various experimental and analytical results
illustrating universality, with particular emphasis
on recent work on universality in cyber algorithms. This is joint work at various stages with a number of authors, Christian Pfrang, Govind Menon, Sheehan Olver, Steven Miller, and particularly Tom Trogdon. 

CATW03 
10th December 2019 10:00 to 11:00 
Nick Trefethen 
Masterclass: polynomial and rational approximation
Part 2 of a threehour series on Complex Computing in MATLAB and Chebfun


CATW03 
10th December 2019 11:30 to 12:30 
Pavel Lushnikov 
Motion of complex singularities and Hamiltonian integrability of surface dynamics
A motion of fluid's free surface is considered in two dimensional (2D) geometry. A timedependent conformal transformation maps a fluid domain into the lower complex halfplane of a new spatial variable. The fluid dynamics is fully characterized by the complex singularities in the upper complex halfplane of the conformal map and the complex velocity. Both a single ideal fluid dynamics (corresponds e.g. to oceanic waves dynamics) and a dynamics of superfluid Helium 4 with two fluid components are considered. Both systems share the same type of the noncanonical Hamiltonian structure. A superfluid Helium case is shown to be completely integrable for the zero gravity and surface tension limit with the exact reduction to the Laplace growth equation which is completely integrable through the connection to the dispersionless limit of the integrable Toda hierarchy and existence of the infinite set of complex pole solutions. A single fluid case with nonzero gravity and surface tension turns more complicated with the infinite set of new moving poles solutions found which are however unavoidably coupled with the emerging moving branch points in the upper halfplane. Residues of poles are the constants of motion. These constants commute with each other in the sense of underlying noncanonical Hamiltonian dynamics. It suggests that the existence of these extra constants of motion provides an argument in support of the conjecture of complete Hamiltonian integrability of 2D free surface hydrodynamics. 

CATW03 
10th December 2019 13:30 to 14:00 
Yuji Nakatsukasa 
Vandermonde with Arnoldi
Vandermonde matrices are exponentially illconditioned, rendering the familiar “polyval(polyfit)” algorithm for polynomial interpolation and leastsquares fitting ineffective at higher degrees. We show that Arnoldi orthogonalization fixes the problem.


CATW03 
10th December 2019 14:00 to 14:30 
Andrew Horning 
Computing spectral measures of differential and integral operators
Unlike its matrix counterpart, the spectral measure of a selfadjoint operator may have an absolutely continuous component and an associated density function, e.g., in applications posed on unbounded domains. The stateoftheart computational methods for these problems typically approximate the density function using a smoothed sum of Dirac measures, corresponding to the spectral measure of a matrix discretization of the operator. However, it is often difficult to determine the smoothing and discretization parameters that are necessary to accurately and efficiently resolve the density function. In this talk, we present an adaptive framework for computing the spectral measure of a selfadjoint operator that provides insight into the selection of smoothing and discretization parameters. We show how to construct local approximations to the density that converge rapidly when the density function is smooth and discuss possible connections with Pade approximation that could alleviate deteriorating convergence rates near nonsmooth points.


CATW03 
10th December 2019 14:30 to 15:00 
Abinand Gopal 
Solving the Laplace and Helmholtz equation on domains with corners using rational functions and their analogs
The solutions to elliptic PDEs on domains with corners often exhibit singularities at the corners, which pose difficulties for numerical methods. Recently, it was found that the approximation power of rational functions can be used in this context to obtain higherorder convergence, in a manner related to the method of fundamental solutions. In this talk, we focus on the extension of these ideas to the case of Helmholtz equation.


CATW03 
10th December 2019 15:00 to 15:30 
Marcus Webb 
Approximation Theory for a Rational Orthogonal Basis on the Real Line The MalmquistTakenaka basis is a rational orthogonal basis constructed by mapping the Laurent basis from the unit circle to the real line by a Möbius transformation and multiplying by a weight to ensure orthogonality. Over the last century its properties have piqued the interest of various researchers including Boyd, Weideman, Christov, and Wiener. Despite this history, the approximation theory of this basis still defies straightforward description. For example, it was shown by Boyd and Weideman that for entire functions the convergence of approximation is superalgebraic, but that exponential convergence is only possible if the function is analytic at infinity (i.e. at the top of the Riemann sphere  quite a strong condition). Nonetheless, convergence can be surprisingly quick, and the main body of this talk will be the result that wave packets clearly cannot have exponentially convergent approximations, but they /initially/ exhibit exponential convergence for large wave packet frequencies with exponential convergence rate proportional to said frequency. Hence, O(log(eps) omega) coefficients are required to resolve a wave packet to an error of O(eps). The proof is by the method of steepest descent in the complex plane. This is joint work with Arieh Iserles and Karen Luong (Cambridge). 

CATW03 
10th December 2019 16:00 to 16:30 
Mikael Slevinsky 
Fast and stable rational approximation of generalized hypergeometric functions Generalized hypergeometric functions are a central tool in the theory of special functions and complex analysis. Rational approximation permits analytic continuation of formal power series well beyond their radii of convergence. In this talk, we will use sequence transformations to convert successive partial sums of generalized hypergeometric series into rational approximants, which are in certain cases Padé approximants. We will describe algorithms for their computation in linear time. This improvement, over traditional algorithms with quadratic complexity, also increases their numerical stability and offers insight into the localization of their poles. 

CATW03 
10th December 2019 16:30 to 17:00 
Birgit Schörkhubre 
Blowup for the supercritical cubic wave equation
This talk is concerned with recent results on singularity formation for the focussing cubic wave equation. We found that in the energy supercritical regime this equation admits an explicit selfsimilar blowup solution, which is stable along a codimension one manifold. Furthermore, based on numerical experiments, we conjecture that this manifold is a threshold between finitetime blowup and dispersion. This is joint work with Irfan Glogić and Maciej Maliborski (University of Vienna).


CATW03 
11th December 2019 09:00 to 10:00 
Sheehan Olver, Thomas Trogdon 
Masterclass: singular integrals and orthogonal polynomials
Orthogonal polynomials are fundamental tools in numerical methods, including for numerical methods for singular integral equations. A known result is that Cauchy transforms of weighted orthogonal polynomials satisfy the same threeterm recurrences as the orthogonal polynomials themselves for n > 0. This basic fact leads to extremely effective schemes of calculating singular integrals that converge spectrally fast (faster than any algebraic power), uniformly in the complex plane. Closed formulae for Cauchy transforms on more complicated geometries are derivable using the Plemelj lemma. These techniques extend to other singular integrals such as those with logarithmic kernels. We will demonstrate these results in Julia using ApproxFun.jl and SingularIntegralEquations.jl. 

CATW03 
11th December 2019 10:00 to 11:00 
Nick Trefethen 
Masterclass: conformal mapping
Part 3 of a threehour series on Complex Computing in MATLAB and Chebfun


CATW03 
11th December 2019 11:30 to 12:30 
Folkmar Bornemann 
Numerical Generatingfunctionology: Counting with Toeplitz Determinants, HaymanAdmissibility, and the WienerHopfFactorization
Counting related to representation theory and symmetric functions can be framed as generating functions given by Toeplitz determinants. Prime examples are counting all permutations with no long increasing subsequence or lattice paths in last passage percolation. Intricate scaling limits of those generating functions have been used, e.g., in the seminal work by Baik/Deift/Johansson, to obtain asymptotic formulae in terms of random matrix theory. In this talk, we address the question whether generating functions can be used to numerically extract the counts in a mesoscopic regime where combinatorial methods are already infeasible and the random matrix asymptotics is still too inaccurate. The stable computation of the counts by means of complex analysis is possible, indeed, and can be explained by the theory of Hayman admissibility. As a bonus track from complex analysis, the numerical evaluation of the Toeplitz determinant itself has to be stabilized by a variant of the BorodinOkounkov formula based on the WienerHopf factorization. This way, we obtain, e.g., exact 1135digit counts in permutations of order 500 or, by taking Hayman’s famous generalization of Stirling’s formula at face value, a blazingly fast, surprisingly robust and accurate numerical asymptotics.


CATW03 
12th December 2019 09:00 to 10:00 
Sheehan Olver, Thomas Trogdon 
Masterclass: Riemann–Hilbert problems
Riemann–Hilbert problems are complex analytical problems where a jump is specified on a complicated contour, often with multiple selfintersections and open endpoints. The numerical methods for singular integrals in Part I lead naturally to an effective collocation scheme provided extra care is taken at selfintersections to ensure the solution is sectionally analytic and smoothness is preserved. Applications discussed include special functions, integrable PDEs, computing orthogonal polynomials, and computing random matrix statistics. We will demonstrate these results in Julia using RiemannHilbert.jl. 

CATW03 
12th December 2019 10:00 to 11:00 
Robert Corless 
Masterclass: The Computer Algebra System Maple, in 2019
I'm tempted to advertise this as "Not Your Parent's Maple", but on the offchance that Pedro Gonnet drops in (or watches the recording), I won't. Maple is a computer algebra system and a PSE (Problem Solving Environment) and has been in wide use since its invention in the early 1980's. A lot has changed since then. In this first hour I will give an overview: Getting Started, What you can and can't do in Computer Algebra; Linear Algebra (actually I am going to "punt" that and let you read our chapter in the CRC Handbook of Linear Algebra, Leslie Hogben ed, 2nd ed; there is an online copy you can read in the Betty and Gordon Moore Library); and choose from about a dozen other topics including integration, differential equations, solving multivariate polynomial systems, and a listing of the current packages in Maple (more than a hundred and forty of them). 

CATW03 
12th December 2019 11:30 to 12:30 
Maria LopezFernandez 
Efficient algorithms for convolutions based on contour integral methods
We propose an efficient family of algorithms for the approximation of Volterra integral equations of convolution type arising in two different applications. The first application we consider is the approximation of the fractional integral and associated fractional differential equations. The second application is the resolution of Schrödinger equations with concentrated potentials, which admit a formulation as systems of integral equations. In both cases, we are able to derive fast implementations of Lubich's Convolution Quadrature with very much reduced memory requirements and very easy to implement. Our algorithms are based on special contour integral representations of the Convolution Quadrature weights, according to the application, and special quadratures to compute them. Numerical experiments showing the performance of our methods will be shown. This is joint work with Lehel Banjai (HeriotWatt University) 

CATW03 
12th December 2019 13:30 to 14:00 
Ted Johnson 
Vortical sources on the walls of rotating containers: a model for oceanic outflows
Fluid of uniform vorticity is expelled from a line source
against a wall. An exact analytical solution is obtained for the nonlinear
problem determining the final steady state. Sufficiently close to the source,
the flow is irrotational and isotropic, turning on the vortical scale
$\sqrt{Q/\omega}$ (for area flux $Q$ and vorticity $\omega$) to travel along
the wall to the right (for $\omega>0$). The flow is linearly stable with
perturbations propagating unattenuated along the interface between vortical and
irrotational fluid. Fully nonlinear numerical integrations of the timedependent
equations of motion show that flow started from rest does indeed closely
approach the steady state.
Superposing an opposing dipole at the origin changes the
momentum flux of the flow and leads to the growth of a bulge in the flow near
the origin, a phenomenon seen in many rotating outflow experiments. When
density variations are allowed the governing equation is no longer Laplace's
equation but solutions can be obtained through a longwave theory. Coauthors: Sean Jamshidi (UCL), Robb McDonald (UCL) 

CATW03 
12th December 2019 14:00 to 14:30 
Sonia Mogilevskaya 
Asymptotic modeling of composite materials with thin coatings by using complex variables
CoAuthors: Svetlana Baranova (University of Minnesota) Dominik Schillinger and Hoa Nguyen (Leibniz Universität Hannover�) Recent advances in surface chemistry made it possible to create materials with ultrathin highperformance coating layers. Numerical modeling of such structures is a challenging task, as accurate resolution of thin layers with standard continuumbased numerical methods, e.g. FEM or BEM, would require prohibitively fine mesh sizes. To avoid this, it has been proposed in the literature to replace a finitethickness coating layer by an interface of zero thickness and model the associated jump conditions in the relevant fields. The existing models, however, are low order accurate with respect to the thickness of the layer. The reasons for this considerable limitation are related to theoretical difficulties in constructing accurate higherorder interface models and to computational difficulties in integrating these models into standard FEM formulations characterized by low regularity conditions for the involved fields and geometry. This talk presents a) a new complex variables based approach in developing arbitrary orders interface models for twodimensional potential problems involving thin isotropic interphase layers and b) a new variationally consistent FEM discretization framework to naturally deal with higherorder derivatives on complex surfaces. Theoretical and computational benefits of the proposed approach will be discussed. 

CATW03 
12th December 2019 14:30 to 15:00 
Anna Zemlyanova 
Fracture mechanics with SteigmannOgden surface energy
It is known that surface energy plays an increasingly important role in modeling of objects at nanoscales. In this talk, surface energy model proposed by Steigmann and Ogden will be applied to fracture problems in twodimensional setting. Introduction of surface energy leads to nonclassical boundary conditions incorporating both stresses and derivatives of displacements. The mechanical problem will be reduced to systems of singular integrodifferential equations which are further reduced to systems of weaklysingular equations. The existence and uniqueness of the solution will be discussed. It is shown that taking into account surface mechanics leads to the sizedependent solutions. A numerical scheme of the solution of the systems of singular integrodifferential equations based on Bsplines is suggested, and the numerical results are presented for different values of the mechanical and the geometric parameters.


CATW03 
12th December 2019 15:00 to 15:30 
Irina Mitrea 
Validated Numerics Techniques for Singular Integrals In this talk I will discuss spectral properties of singular 

CATW03 
12th December 2019 16:00 to 16:30 
Shari Moskow 
Reduced order models for spectral domain inversion: embedding into the continuous problem and generation of internal data.*
We generate datadriven reduced order models (ROMs) for inversion of the one and two dimensional Schr\"odinger equation in the spectral domain given boundary data at a few frequencies. The ROM is the Galerkin projection of the Schr\"odinger operator onto the space spanned by solutions at these sample frequencies, and corresponds to a rational interpolant of the Neumann to Dirichlet map. The ROM matrix is in general full, and not good for extracting the potential. However, using an orthogonal change of basis via Lanczos iteration, we can transform the ROM to a block triadiagonal form from which it is easier to extract the unknown coefficient. In one dimension, the tridiagonal matrix corresponds to a threepoint staggered finitedifference system for the Schr\"odinger operator discretized on a socalled spectrally matched grid which is almost independent of the medium. In higher dimensions, the orthogonalized basis functions play the role of the grid steps. The orthogonalized basis functions are localized and also depend only very weakly on the medium, and thus by embedding into the continuous problem, the reduced order model yields highly accurate internal solutions. That is to say, we can obtain, just from boundary data, very good approximations of the solution of the Schr\"odinger equation in the whole domain for a spectral interval that includes the sample frequencies. We present inversion experiments based on the internal solutions in one and two dimensions. * joint with L. Borcea, V. Druskin, A. Mamonov, M. Zaslavsky 

CATW03 
12th December 2019 16:30 to 17:00 
Manfred Trummer 
The Szegö Kernel and Oblique Projections: Conformal Mapping of Nonsmooth Regions
A method for computing the Riemann mapping function of a smooth domain is extended to include the case of simply connected convex regions with corners, in particular convex polygons. The method expresses the Szegö kernel as the solution of an integral equation; the equation is modified to allow for corners in the region. 

CATW03 
13th December 2019 09:00 to 10:00 
Sheehan Olver, Thomas Trogdon 
Masterclass: oscillatory RiemannHilbert problems
RiemannHilbert problems arising in applications are often oscillatory presenting challenges to their numerical solution. An effective scheme for determining their asymptotic behaviour is DeiftZhou steepest descent, which mirrors steepest descent for oscillatory integrals by deforming to paths that turn oscillations to exponential decay. This is a fundamental result that lead to numerous important rigorous asymptotic results over the last 35+ years. This technique proves useful for numerics as well providing a convergent approach that is accurate both in the asymptotic and nonasymptotic regime. Recent progress on going beyond steepest descent and solving oscillatory problems without deformation using GMRES is also discussed.


CATW03 
13th December 2019 10:00 to 11:00 
Robert Corless 
Masterclass: Programming in Maple: an extended example using Bohemians A Bohemian is a BOunded HEight Matrix of Integers (BOHEMI, close enough). Recently I have become very interested in such things; see bohemianmatrices.com for some reasons why. In this hour we will look at a collection of Maple procedures designed (this month!) to answer questions about complex symmetric, tridiagonal, irreducible, zerodiagonal Bohemians (a new class, chosen just for this workshop). This means that there will be (#P)^(m1) mdimensional such matrices for a given population of elements P, which for this class cannot contain zero. We will look at fast ways to generate these matrices, how to generate fast(ish) code to compute the characteristic polynomials (and why), and generally use this topic as an excuse to learn some Maple programming. The hour will assume some familiarity with programming; for instance, if you know Matlab, then you very nearly know Maple already (in some ways they are similar enough that it causes confusion, unfortunately). But it will not be necessary; I hope to encourage a friendly atmosphere and we'll generate some interesting (I hope) images, and perhaps some interesting mathematical conjectures. But even if you know Maple well, you might learn something interesting. All the scripts/worksheets/workbooks have been made available at http://publish.uwo.ca/~rcorless/Maple2019/ so you may download them and run and modify the examples yourself, and generate your own Bohemian images. Indeed I believe that it is entirely likely that you will be able to formulate your own Bohemian conjectures during this activity; and it has been known for participants to prove theorems about them, during the lecture. Who knows, perhaps your next paper will get its main result during this activity. Licences for Maple valid for one month have been generously provided for participants by Maplesoft. There will be a representative from Maplesoft here to answer any questions. 

CATW03 
13th December 2019 11:30 to 12:30 
Christian Klein 
Computational approach to compact Riemann surfaces
A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a twodimensional Newton iteration. The starting values for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros. A set of generators of the fundamental group for the complement of these critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the de ning equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing ClenshawCurtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the ClenshawCurtis algorithm and contour integrals. A special approach is presented for hyperelliptic curves in Weierstrass normal form.


CATW03 
13th December 2019 13:30 to 14:00 
Bernard Deconinck 
The numerical solution of BVPs using the Unified Transform Method
Using a semidiscrete analog of the Unified Transform
Method of Fokas, we explore the possibility of devising
finitedifference
schemes that take boundary conditions into account
properly. This is done by
combining the exact solution for the linear problem with
the solution of the
nonlinear problem in a splitstep approach. In this talk,
I will discuss the
first step of this project, where the solution of the
linear problem is
explored. Coauthor: Jorge Cisneros (University of Washington) 

CATW03 
13th December 2019 14:00 to 14:30 
Tom DeLillo 
Remarks on numerical methods for conformal mapping of multiply connected domains and applications Conformal maps from multiply connected domains with circular boundaries to physical domains and complex velocity potentials for potential flow problems can be represented by Laurent series in the circle domains. The linear systems arising in the computation of the truncated series have a block structure in the form of the discrete Fourier transform plus low rank matrices representing the interaction of the circles plus certain auxiliary 

CATW03 
13th December 2019 14:30 to 15:00 
Olivier Sète 
Computing Walsh's conformal map onto lemniscatic domains
The conformal mapping of multiply connected domains onto lemniscatic domains was introduced by Walsh in 1956. Walsh's map is a direct generalization of the Riemann mapping. Several important quantities, such as the Green's function with pole at infinity of the multiply connected domain, or the logarithmic capacity of its complemet, can be read of the lemniscatic domain and Walsh's map. In this talk, we discuss the computation of this conformal mapping and lemniscatic domain. This is based on joint work with Jörg Liesen and Mohamed Nasser.


CATW03 
13th December 2019 15:00 to 15:30 
Peter Baddoo 
Computing periodic conformal mappings
Conformal mappings are used to model a range of phenomena in the physical sciences. Although the Riemann mapping theorem guarantees the existence of a mapping between two conformally equivalent domains, actually constructing these mappings is extremely challenging. Moreover, even when the mapping is known in principle, an efficient representation is not always available. Accordingly, we present techniques for rapidly computing the conformal mapping from a multiply connected canonical circular domain to a periodic array of polygons. The boundary correspondence function is found by solving the parameter problem for a new periodic SchwarzChristoffel formula. We then represent the mapping using rational function approximation. To this end, we present a periodic analogue of the adaptive AntoulasAnderson (AAA) algorithm to obtain the relevant support points and weights. The procedure is extremely fast; evaluating the mappings typically takes around 10 microseconds. Finally, we leverage the new algorithms to solve problems in fluid mechanics in periodic domains.


CATW03 
13th December 2019 16:00 to 16:30 
Alfredo Deaño 
Computational aspects of complex orthogonal polynomials Complex orthogonal polynomials, and associated quantities such as recurrence coefficients, have been recently used in problems related to highly oscillatory quadrature (construction of complex Gaussian quadrature rules), in the analysis of nonHermitian random matrix ensembles and also in the study of solutions of Painlevé differential equations. In this talk we will present examples and propose several options for their numerical evaluation. 

CATW03 
13th December 2019 16:30 to 17:00 
Philippe Trinh 
The search for complex singularities in exponential asymptotics
In the last two decades, the development of specialized techniques in mathematics known as exponential asymptotics has led to the successful resolution of longstanding problems in topics as varied as quantum mechanics, crystal growth, dislocations, pattern formation, turbulence, thin film flow, and hydrodynamics. These developments have emerged from the realization that in many such problems, exponentially small effects, linked to the presence of singularities in the analytic continuation of a perturbative solution, can significantly change the solutions of the underlying mathematical models. However, the majority of problems studied have been onedimensional in nature (essentially ordinary differential or difference equations). The extension of exponential asymptotics to multidimensional problems requires some fundamental advances in our ability to compute numerical analytic continuations of single and multivariate complex functions. In this talk, I shall present the basic theory of exponential asymptotics and discuss these open challenges in the context of problems in fluid mechanics. 

CAT 
17th December 2019 14:00 to 15:00 
Giovani Vasconcelos 
Bubble dynamics and velocity selection in a HeleShaw cell
The unsteady motion of a finite assembly of bubbles in a
HeleShaw channel
is studied in the case when surface tension is neglected. A
general exact
solution is obtained in terms of a conformal map from a
multiply connected
circular domain to the fluid region exterior to the bubbles.
The correspond
ing mapping function is given explicitly in terms of certain
special transcen
dental functions, known as the secondary SchottkyKlein
prime functions.
Exploring the properties of these solutions, we show that
steady configura
tions where the bubbles move with a velocity, U, which is
twice greater than
the velocity, V , of the background flow, i.e., U = 2V , are
the only attrac
tor of the dynamics; whereas solutions with U ≠ 2V act as
repellors. This
demonstrates that the special nature of the solutions with U
= 2V is already
builtin in the zerosurfacetension dynamics, which is
confirmed by the in
clusion of regularization effects. In particular, the case
of a single bubble will
be discussed in detail and several numerical examples of
bubble evolution
and bubble selection will be presented.
