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Seminars (CAT)

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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 Korteweg-deVries, nonlinear Klein-Gordon 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 re-discovered 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 Muttalib-Borodin ensembles
I will present joint work with Christophe Charlier and Julian Mauersberger.
We consider the limiting process that arises at the hard edge of Muttalib-Borodin 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 Riemann-Hilbert 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 water-wave 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 Dirichlet-to-Neumann operator over a polygonal bottom profile. One very recent example applies to a hydrodynamic pilot-wave 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 non-Hermitian random matrix ensembles
In this talk we present recent results on the connection between Painlevé equations and NxN non-Hermitian 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 Riemann-zeta function on the critical line Re(z) =1/2. Amongst many others, there is random matrix theory, the asymptotic theory of orthogonal polynomials, self-similar 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 one-parameter 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 semi-classical 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 q-lattice
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 Pseudo-Jacobi 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 non-generic 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 Riemann-zeta function on the critical line Re(z) =1/2. Amongst many others, there is random matrix theory, the asymptotic theory of orthogonal polynomials, self-similar 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 one-parameter 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 semi-classical 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 Riemann-Hilbert 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 non-trivial 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.

Co-authors include: T. Anselmo, J.-J. Barragán-Amado, 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 Riemann-Hilbert problem set on a union of non-intersecting smooth closed curves with generic jump matrix. The main focus will be on the one-circle 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 Nekrasov-Okounkov instanton partition functions for Riemann-Hilbert 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 Riemann-zeta function on the critical line Re(z) =1/2. Amongst many others, there is random matrix theory, the asymptotic theory of orthogonal polynomials, self-similar 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 one-parameter 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 semi-classical 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 two-dimensional 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 Levi-Civita 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 low-Reynolds-number swimming, we consider viscous flows in two-dimensional channels and present new transform methods for analysing such problems. The new methods provide a unified general approach to finding quasi-analytical solutions to a wide range of problems in low-Reynolds-number hydrodynamics and plane elasticity. In this talk, we focus on pressure-driven 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 rank-one 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 Riemann-Hilbert 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 Stuart-type vorticity
A new family of exact solutions to the two-dimensional 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 non-zero vorticity of “Stuart-type”. 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 Stuart-embedded point vortex equilibria which results in a two-real-parameter 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 so-called inverse magnetization problem in paleomagnetic context. In such a problem the aim is to recover the average remaneWe consider so-called 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 complex-analysis 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 two-dimensional 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 finite-time 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) 187-223. [2] M.M.S. Nasser, Fast solution of boundary integral equations with the generalized Neumann kernel, Electron. Trans. Numer. Anal. 44 (2015) 189-229.
CATW01 13th September 2019
10:00 to 11:00
Lesley Ward The harmonic-measure distribution function of a planar domain, and the Schottky-Klein prime function
The $h$-function or harmonic-measure 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 Schottky-Klein 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 Riemann-Hilbert Problems
Nonlinear Riemann-Hilbert 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 Riemann-Hilbert 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 Riemann-Hilbert problems and a
class of extremal problems is established. Solutions to Riemann-Hilbert
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 anti-analytic part.
CAT 24th September 2019
14:00 to 15:00
Elias Wegert Nonlinear Riemann-Hilbert Problems (continued)
Nonlinear Riemann-Hilbert 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 Riemann-Hilbert 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 Riemann-Hilbert problems and a
class of extremal problems is established. Solutions to Riemann-Hilbert
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 anti-analytic 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 time-dependent
conformal mapping of a fluid domain into the lower complex half-plane. We
reformulate the exact Eulerian dynamics through a non-canonical nonlocal
Hamiltonian system for the pair of new conformal variables. The
corresponding non-canonical Poisson bracket is non-degenerate, 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 non-canonical Hamiltonian structure. In both cases the fluid
dynamics is fully characterized by the complex singularities in the upper
complex half-plane 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 non-canonical 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 multi-variable 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 multi-variable complex analysis?   In this talk we provide a broad overview of the basic features of several complex variables with an eye towards building a ``multi-variable complex analysis toolbox’’.



 

CAT 8th October 2019
14:00 to 15:00
Helena Stage A heuristic introduction to the applications of Wiener-Hopf 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 Wiener-Hopf 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 non-linear 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 non-linear 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, 
Camassa-Holm, Hunter-Saxton, and KdV.





CAT 17th October 2019
16:00 to 17:00
Matthew Turner Time-dependent conformal mapping techniques applied to fluid sloshing problems

In this presentation we demonstrate how time-dependent conformal mappings can be used to construct a fast and effective numerical scheme for examining two-dimensional, inviscid, irrotational fluid sloshing in both fixed and moving vessels. In particular we examine how horizontal side-wall baffles can be incorporated into the system creating a multiply-connected fluid domain.



 

CAT 24th October 2019
10:30 to 12:00
Saleh Tanveer Free-boundary problems, singularities and exponential asymptotics
CAT 24th October 2019
14:00 to 15:30
Alexander Its The Riemann-Hilbert method. Toeplitz determinants as a case study
The Riemann-Hilbert 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 Wiener-Hopf method. In its current form, the Riemann-Hilbert
approach exploits ideas which goes beyond the usual Wiener-Hopf 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 Riemann-Hilbert method, is the global asymptotic analysis of nonlinear
systems. Indeed, many long-standing asymptotic problems in the diverse areas of
pure and applied math have been solved with the help of the Riemann-Hilbert
technique.
One of the recent applications of the Riemann-Hilbert 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 Riemann-Hilbert 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 RIemann-Hilbert 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\'on-Zygmund theory on Uniformly Rectifiable sets, a sharp divergence theorem with non-tangential boundary traces, Fatou-type 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 Free-boundary problems, singularities and exponential asymptotics
CAT 25th October 2019
14:00 to 15:30
Alexander Its The Riemann-Hilbert method. Toeplitz determinants as a case study
The Riemann-Hilbert 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 Wiener-Hopf method. In its current form, the Riemann-Hilbert
approach exploits ideas which goes beyond the usual Wiener-Hopf 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 Riemann-Hilbert method, is the global asymptotic analysis of nonlinear
systems. Indeed, many long-standing asymptotic problems in the diverse areas of
pure and applied math have been solved with the help of the Riemann-Hilbert
technique.
One of the recent applications of the Riemann-Hilbert 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 Riemann-Hilbert 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 RIemann-Hilbert 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 semi-infinite Neo-Hookean 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 Complex-plane 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 so-called lubricant-infused surfaces (SLIPS). This talk addresses the flow phenomena close to such surfaces. The nature of the structured surface leads to a mixed-boundary 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 Riemann-Hilbert problems and applications
It is hoped that 1/2 of this presentation will be introductory, explaining with examples how Riemann-Hilbert 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 Hele-Shaw cell

In a traditional Hele-Shaw configuration, the governing equation for the pressure is Laplace's equation; thus, mathematical models for Hele-Shaw flows are amenable to complex analysis.  We consider here one such problem, where a bubble is moving steadily in a Hele-Shaw cell.  This is like the classical Taylor-Saffman 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 well-studied finger and bubble problems in a Hele-Shaw 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 well-known selection problems in Hele-Shaw 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 small-amplitude 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 Martinez-Finkelshtein 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 vector-valued 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 non-trivial, 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 Riemann-Hilbert approach
Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distribution in models of positive-temperature free fermions. I will explain how logarithmic derivatives of the Fredholm determinants can be expressed in terms of a $2\times 2$ Riemann-Hilbert 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 Cauchy-type integrals in multivariable complex analysis

This is joint work with Elias M. Stein (Princeton University).

The classical Cauchy theorem and Cauchy integral formula for analytic functions of one complex variable give rise to a plethora of applications to Physics and Engineering, and as such are essential components of the Complex Analysis Toolbox.

Two crucial features of the integration kernel of the Cauchy integral (Cauchy kernel, for short) are its ``analyticity’’ (the Cauchy kernel is an analytic function of the output variable) and its ``universality’’ (the Cauchy integral is meaningful for almost any contour shape). One drawback of the Cauchy kernel is that it lacks a good transformation law under conformal maps (with a few exceptions).

This brings up two questions: Are there other integration kernels that retain the main features of the Cauchy kernel but also have good transformation laws under conformal maps? And: is there an analog of the Cauchy kernel for analytic functions of two (or more) complex variables that retains the aforementioned crucial features?

In this talk I will give a survey of what is known of these matters, with an eye towards enriching the Complex Analysis Toolbox as we know it, and towards building a ``Multivariable Complex Analysis Toolbox’’.

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 Kadomtsev-Petviashvili I (KPI) equation, which are all soliton equations solvable by the inverse scattering.

The Boussinesq equation was introduced by Boussinesq in 1871 to describe the propagation of long waves in shallow water. Rational solutions of the Boussinesq equation, which are algebraically decaying and depend on two arbitrary parameters, are expressed in terms of special polynomials that are derived through a bilinear equation, have a similar appearance to rogue-wave solutions of the focusing NLS equation and have an interesting structure. Conservation laws and integral relations associated with rational solutions of the Boussinesq equation will also be discussed.

Rational solutions of the KPI equation will be derived in three ways: from rational solutions of the NLS equation; from rational solutions of the Boussinesq equation; and from the spectral problem for the KPI equation. It'll be shown that these three families of rational solutions are fundamentally different.

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 well-known 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 no-slip and no-shear 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 Anna-Karin 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 semi-plane. The results of the present work generalize the results for contact problems with Gurtin-Murdoch elasticity by including additional dependency on the curvature of the surface. The mechanical problem is reduced to a system of singular integro-differential equations which is regularized using Fourier transform method. The size-dependency of the solutions of the problem is highlighted. It is observed that the curvature-dependence 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 two-dimensional protrusions in the form of either straight needles or curved fingers satisfying Loewner's equation is studied using the Schwarz-Christoffel (SC) map. Particular use is made of Driscoll's numerical procedure, the SC Toolbox, for computing the SC map from a half-plane to a slit half-plane, 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 half-plane and channel geometries. Bifurcating fingers are also studied and application to branching stream networks discussed.

(ii) Solutions are found for the growth of infinitesimally thin, two-dimensional fingers governed by Poisson's equation in a long strip. The analytical results determine the asymptotic paths selected by the fingers which compare well with the recent numerical results of Cohen and Rothman (2017) for the case of two and three fingers. The generalisation of the method to an arbitrary number of fingers is presented and further results for four finger evolution given. The relation to the analogous problem of finger growth in a Laplacian field is also 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 two-dimensional 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 non-uniqueness of Heine-Stieltjes polynomials and to the asymptotics of some non-classical 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 multi-phase 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 stress-strain fields are calculated and the averaged elastic constants are obtained in symbolic form. Extensions of Maxwell's approach and other various self-consisting 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 streamline-vorticity 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 non-constant 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 stream-function. 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 Riemann-Hilbert 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 right-angled 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 order-2 vector Riemann-Hilbert problems. When the wedge is concave (a convex obstacle), the acoustic problem is transformed into an order-3 Riemann-Hilbert problem. The order-2 and 3 vector Riemann-Hilbert problems are solved by recasting them as scalar Riemann-Hilbert 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 Wiener-Hopf 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 time-dependent gap Hele-Shaw cell

Hele-Shaw 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 Hele-Shaw 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 Riemann-Hilbert problem and an abstract Cauchy-Kovalevsky 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 most-studied special functions; see in particular Philip J. Davis' Chauvenet prize-winning 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) Initial-boundary 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 t-asymptotics 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 t-Periodic Data: I. Exact Results, Proc. R. Soc. A 471, 20140925
(2015).
J. Lenells and A. S. Fokas, The Nonlinear Schrödinger Equation with
t-Periodic 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 Two-Parameter 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) Initial-boundary 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 t-asymptotics 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 t-Periodic Data: I. Exact Results, Proc. R. Soc. A 471, 20140925
(2015).
J. Lenells and A. S. Fokas, The Nonlinear Schrödinger Equation with
t-Periodic 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 Two-Parameter 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 Hele-Shaw flow
When a less viscous fluid is injected into a more viscous fluid in a Hele-Shaw, the interface between the fluids is unstable.  This is the Saffman-Taylor instability which can give rise to a striking pattern formation at the interface.  The usual model for this type of flow is a two-dimensional moving boundary problem which is amenable to complex variable methods, especially in the special case in which surface tension is ignored.  For non-standard geometries progress can be made via numerical simulation, using a level set formulation for example.  Some recent experimental, analytical and numerical studies of non-standard moving boundary problems in Hele-Shaw 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 Riemann-Hilbert problems of the theory of automorphic functions and inverse problems of elasticity and cavitating flow for multiply connected domains
The general theory of the Riemann-Hilbert problem for piece-wise 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 n-connected circular domain into the physical domain are reconstructed by solving two Riemann-Hilbert problems of the theory of piece-wise 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 thin-walled 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 fixed-angle corner at the trailing edge is introduced by way of a Karman-Trefftz 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 three-hour 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 trade-off 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 multi-sheeted, 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 well-understood. For example, it is well-known 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

Co-Authors: Daan Huybrechs, David Hewett

When modelling high frequency scattering, a common approach is to enrich the approximation space with oscillatory basis functions. This can lead to a significant reduction in the DOFs required to accurately represent the solution, which is advantageous in terms of memory requirements and it makes the discrete system significantly easier to solve. A potential drawback is that the each element in the discrete system is a highly oscillatory, and sometimes singular, integral. Therefore an efficient quadrature rule for such integrals is essential for an efficient scattering model. In this talk I will present a new class of quadrature rule we have designed for this purpose, combining Numerical Steepest Descent (which works well for oscillatory integrals) with Generalised Gaussian quadrature (which works well for singular integrals).

CATW03 9th December 2019
15:00 to 15:30
Matthew Colbrook The Foundations of Infinite-Dimensional 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 half-century. However, such computations are infamously difficult, since standard approaches do not, in general, produce correct solutions (the most famous problem in the self-adjoint 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 Attouch-Wets 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 long-standing 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 (projection-valued) spectral measures of self-adjoint 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.

Firstly, I will talk about the situation when boundary conditions are given on many plates and the associated Wiener-Hopf equation. The Wiener-Hopf method can be considered as a Riemann-Hilbert method with added analytisity information allowing for full power of complex analysis methods to be exploited. I will explain the difficulties involved when matrix Wiener-Hopf equations are solved. I will describe how recent advances in computation of rational approximations and orthogonal polynomials can be used. The corresponding canonical acoustics scattering problems will be considered.
This is joint work with Matthew J. Priddin and Lorna J. Ayton.

Secondly, I will talk about how special functions called Mathieu function can be used to solve Helmholz equation with boundary conditions given on a plate. This allows to produce a fast and stable code for challenging conditions like varying elasticity and porosity on a plate. This is joint work with Matthew J. Colbrook.

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 far-field behaviour of the solutions can be exploited to construct tailor-made 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 semi-infinite 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 d-bar 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 Davey-Stewartson system [2] and the 3-wave 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 3-Wave 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 three-hour 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 time-dependent conformal transformation maps a
fluid domain into the lower complex half-plane of a new spatial
variable. The fluid dynamics is fully characterized by the complex
singularities in the upper complex half-plane 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 non-canonical 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 half-plane.
Residues of poles are the constants of motion. These constants
commute with each other in the sense of underlying non-canonical
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 ill-conditioned, rendering the familiar “polyval(polyfit)” algorithm for polynomial interpolation and least-squares 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 self-adjoint operator may have an absolutely continuous component and an associated density function, e.g., in applications posed on unbounded domains. The state-of-the-art 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 self-adjoint 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 non-smooth 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 higher-order 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 Malmquist-Takenaka 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 self-similar blowup solution, which is stable along a codimension one manifold. Furthermore, based on numerical experiments, we conjecture that this manifold is a threshold between finite-time 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 three-term 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 three-hour 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, Hayman-Admissibility, and the Wiener-Hopf-Factorization
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 Borodin-Okounkov formula based on the Wiener-Hopf factorization. This way, we obtain, e.g., exact 1135-digit 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 self-intersections 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 self-intersections 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 off-chance 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 Lopez-Fernandez 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 (Heriot-Watt 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 time-dependent 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 long-wave theory.

Co-authors: 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
Co-Authors: 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 high-performance coating layers. Numerical modeling of such structures is a challenging task, as accurate resolution of thin layers with standard continuum-based 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 finite-thickness 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 higher-order 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 two-dimensional potential problems involving thin isotropic interphase layers and b) a new variationally consistent FEM discretization framework to naturally deal with higher-order 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 Steigmann-Ogden surface energy
It is known that surface energy plays an increasingly important role in modeling of objects at nano-scales. In this talk, surface energy model proposed by Steigmann and Ogden will be applied to fracture problems in two-dimensional setting. Introduction of surface energy leads to non-classical boundary conditions incorporating both stresses and derivatives of displacements. The mechanical problem will be reduced to systems of singular integro-differential equations which are further reduced to systems of weakly-singular equations. The existence and uniqueness of the solution will be discussed. It is shown that taking into account surface mechanics leads to the size-dependent solutions. A numerical scheme of the solution of the systems of singular integro-differential equations based on B-splines 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
integral operators of layer potential type in the class of curvilinear
polygons in two dimensions and illustrate how a combination of
Harmonic Analysis techniques and Validated Numerics methods
can be successfully implemented for establishing well-posedness
results for boundary value problems for second order elliptic partial
differential operators with constant coefficients in this geometric
setting. This is based on joint work with H. Awala, T. Johnson,
K. Ott and W. Tucker.

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 data-driven 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 three-point staggered finite-difference system for the Schr\"odinger operator discretized on a so-called 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 Non-smooth 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 Riemann-Hilbert problems
Riemann-Hilbert problems arising in applications are often oscillatory presenting challenges to their numerical solution. An effective scheme for determining their asymptotic behaviour is Deift-Zhou 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 non-asymptotic 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, zero-diagonal Bohemians (a new class, chosen just for this workshop). This means that there will be (#P)^(m-1) m-dimensional 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 two-dimensional 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 Clenshaw-Curtis 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 Clenshaw-Curtis 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 semi-discrete analog of the Unified Transform Method of Fokas, we explore the possibility of devising finite-difference 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 split-step approach. In this talk, I will discuss the first step of this project, where the solution of the linear problem is explored.

Co-author: 
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
parameters such as the conformal moduli or the circulations of the flow. The conjugate gradient method can be used to solve these systems efficiently. We will give an outline of this approach. Time permitting, we will also give several examples of conformal map calculations.

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 Schwarz--Christoffel formula. We then represent the mapping using rational function approximation. To this end, we present a periodic analogue of the adaptive Antoulas--Anderson (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 non-Hermitian 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 long-standing 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 one-dimensional in nature (essentially ordinary differential or difference equations). The extension of exponential asymptotics to multi-dimensional problems requires some fundamental advances in our ability to compute numerical analytic continuations of single- and multi-variate 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 Hele-Shaw cell
The unsteady motion of a finite assembly of bubbles in a Hele-Shaw 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 Schottky-Klein 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 built-in in the zero-surface-tension 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.




University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons