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Timetable (CATW02)

Complex analysis in mathematical physics and applications

Monday 28th October 2019 to Friday 1st November 2019

Monday 28th October 2019
09:30 to 09:50 Registration
09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director)
10:00 to 11:00 Martine Ben Amar (CNRS - Ecole Normale Superieure Paris); (Université Pierre et Marie Curie Paris)
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.
11:00 to 11:30 Tea & coffee
11:30 to 12:30 John King (University of Nottingham)
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.
12:30 to 13:30 Lunch at Churchill College
13:30 to 14:00 Clarissa Schoenecker (Technische Universität Kaiserslautern); (Max-Planck-Institut für Polymerforschung)
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).
14:00 to 14:30 Oleg Lisovyy (Université François-Rabelais Tours)
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.
14:30 to 15:30 Ken McLaughlin (Colorado State University)
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.
15:30 to 16:00 Tea & coffee
16:00 to 17:00 Dave Smith (National University of Singapore)
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.
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 29th October 2019
09:00 to 10:00 Scott McCue (Queensland University of Technology)
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.

10:00 to 11:00 Nick Moore (Florida State University)
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.
11:00 to 11:30 Tea & coffee
11:30 to 12:30 Andrei Martinez-Finkelshtein (Baylor University); (University of Almeria)
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.
12:30 to 13:30 Lunch at Churchill College
13:30 to 14:30 Maxim Yattselev (Indiana University-Purdue University Indianapolis)
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.
14:30 to 15:30 Tom Claeys (Université Catholique de Louvain)
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.
15:30 to 16:00 Tea & coffee
16:00 to 17:00 Loredana Lanzani (Syracuse University)
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’’.

Wednesday 30th October 2019
09:00 to 18:00 Industrial Applications of Complex Analysis (Open for business day)
Thursday 31st October 2019
09:00 to 10:00 Peter Clarkson (University of Kent)
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.

10:00 to 11:00 Igor Krasovsky (Imperial College London)
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.
11:00 to 11:30 Tea & coffee
11:30 to 12:30 Michael Siegel (New Jersey Institute of Technology)
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).
12:30 to 13:30 Lunch at Churchill College
13:30 to 14:30 Anna Zemlyanova (Kansas State University)
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.
14:30 to 15:30 Robb McDonald (University College London)
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.

15:30 to 16:00 Tea & coffee
16:00 to 17:00 Kevin O'Neil (The University of Tulsa)
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.

19:30 to 22:00 Formal Dinner at St John's College (Wordsworth Room)
Friday 1st November 2019
09:00 to 10:00 Craig Tracy (University of California, Davis); (University of California, Davis)
Blocks in the asymmetric simple exclusion process
10:00 to 11:00 Vladimir Mityushev (Akademia Pedagogiczna)
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.

11:00 to 11:30 Tea & coffee
11:30 to 12:30 Takashi Sakajo (Kyoto University); (Kyoto University)
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.
12:30 to 13:30 Lunch at Churchill College
13:30 to 14:30 Yuri Antipov (Louisiana State University)
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.
14:30 to 15:30 Gennady Mishuris (Aberystwyth University)
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.
15:30 to 16:00 Tea & coffee
16:00 to 17:00 Xie Xuming (Morgan State University)
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.

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