Modern Applications of Complex Variables: Modeling, Theory and Computation (15w5052)

Arriving in Banff, Alberta Sunday, January 11 and departing Friday January 16, 2015

Organizers

(New Jersey Institute of Technology)

(University of California, San Diego)

(Colorado School of Mines)

(McMaster University)

Objectives

The theme of this workshop can be viewed as touching on three areas: new techniques in complex analysis, computational complex analysis, and modeling by means of complex analysis. There is the potential for exciting and productive mutual exchange between established workers in these areas who would otherwise be unlikely to interact. Participants will emerge with a broader perspective of the field, and will, we expect, forge new collaborations. The five organizers each bring a different, complementary set of mathematical skills and experiences, and between them have assembled a strong international list of participants with diverse but interrelated expertise: the list of proposed speakers is certainly not a reunion of old friends and colleagues. Many of the participants will be meeting for the first time at this workshop. In addition, the workshop will also offer the opportunity for some promising early-career researchers to learn from experts in their own and related fields, and engage in the scheduled discussion sessions.We now give a brief outline of the kinds of problems that we will focus on. The division is somewhat artificial because the workshop themes are all interlinked in various ways: this is precisely why such a forum is needed to bring together mathematicians and scientists with significant knowledge and experience in the relevant fields.Consider a classical topic: the Riemann--Hilbert problem, in which one has to find a function that is analytic everywhere in the complex plane except along a given curve where it has a prescribed jump. Traditionally, such problems arise in the context of singular integral equations and the Wiener--Hopf technique. More recently, Riemann--Hilbert problems have been connected to random matrix theory, nonlinear special functions (Painlev'e transcendents) and nonlinear wave equations. These connections have provoked the development of efficient numerical methods for solving Riemann--Hilbert problems, and these, in turn, will lead back to new methods for problems that can be formulated using the Wiener--Hopf technique.Another classical technique that has undergone a recent dramatic revival is conformal mapping, with the discovery of new exact mappings for multiply connected domains (e.g.~cite{crowdy2005,delillo2004}). The Schwarz function (which has a natural construction in terms of a conformal mapping from a suitable canonical domain) has also found new applications (e.g.~cite{Martin2012}). Conformal mapping, and related techniques of complex analysis, have long been applied to classical two-dimensional free boundary fluid dynamical problems such as the Hele-Shaw problem. Analytic functions have an obvious application in problems such as these, where the underlying flow is governed by the 2D Laplace equation (an additional virtue of these approaches is that the mapping can easily and naturally take account of geometric singularities at corners). The new discoveries in conformal mapping theory have led to a new surge of research activity in such classical problems (e.g.~cite{mccue}), but in addition there is increasing interest in the use of conformal mapping for other governing PDEs: examples are biharmonic problems (in 2D slow flow with free boundaries~cite{crowdy2002}; in quasi-2D slow flow problems that arise in optical fiber manufacture, to describe the evolution of the fiber cross-section~cite{fibres}; and of course in 2D elasticity); two-dimensional boundary integral equations for certain three-dimensional problems; and various forms of the Poisson equation relevant, e.g., to droplets sliding on surfaces~cite{drop1} and to cell locomotion and adhesion and subsequent pattern formation in melanoma~cite{benamar}.Singularities provide another common theme: finding them, classifying them, using them, tracking them as they move. The evolution of singularities characterizing the complex extensions of the solutions to real-valued PDEs offers key information about the regularity of these solutions~cite{Weidman2003}. Complexified forms of the Euler equations have been used to study the finite-time singularity formation in hydrodynamic systems~cite{Pauls2010}; this is closely related to one of the ``millennium problems'' identified by the Clay Mathematics Institute~cite{Fefferman2000}. Free surfaces and interfaces can evolve producing cusped and other near-singular shapes~cite{howison}. Formulations in terms of singularity structure are generally amenable to asymptotic and numerical solution.There have also been new developments with integral transforms. One is the Fokas method for solving linear boundary-value problems~cite{Fokas2008}. Realistic applications of this methodology are in their infancy: there is much scope for further work, and a discussion session devoted to such methods would, we expect, be particularly profitable.Applications of complex variable to numerical analysis are also a topic of burgeoning interest and activity. Conventional numerical methods are quickly running into natural limitations that can only be overcome through the development of new mathematics. Not all problems come posed as ODEs/PDEs/integral equations, ready for application of conventional numerics. Rather, complex variable formulations are common. Examples are conformal mapping, Riemann--Hilbert problems (e.g.~Deift and Zhou's reformulation of perturbations to NLS), or relationships between inverse Cauchy transforms used to calculate free probability operations related to random matrix theory. Even rather ``direct'' numerical methods for ODEs (i.e.~not going via the Riemann--Hilbert machinery) can be formulated so that pole fields actually become the preferred situation over smooth areas~cite{fornberg}.The mathematical techniques that will be discussed should ultimately become part of the arsenal deployed by scientists and engineers in solving real problems in engineering, biological and medical sciences.bibstyle{plainnat} begin{thebibliography}{999}bibitem{drop1} Ben Amar, M., Cummings, L. J., Pomeau, Y. 2003 Transition of a moving contact line from smooth to angular. Phys. Fluids 15, 2949-2960.bibitem{benamar} Chatelain, C., Ciarletta, P., Ben Amar, M. 2011 Morphological changes in early melanoma development: Influence of nutrients, growth inhibitors and cell-adhesion mechanisms. J. Theor. Biol. 290, 46-59.bibitem{crowdy2002} Crowdy, D. G. 2002 Exact solutions for the viscous sintering of multiply-connected fluid domains. J. Eng. 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