For Honours Students Honours Course Modern Asymptotics and Perturbation Theory Projects and Essays Mathematical Immunology Nonlinear Integrable Difference Equations Exponential Asymptotics Honours Course: Modern Asymptotics and Perturbation Theory Differential equations model most natural phenomena we know. Yet their solutions can be notoriously difficult to understand. In place of ``exact'' solutions, a rich array of asymptotic methods have been developed to qualitatively understand the solutions. These are based on either the intrinsic variables or an external parameter in the problem being large or small. The field is vast and ranges from the fundamental theory of asymptotic expansions and perturbation methods, developed by Poincaré for the study of the solar system, to modern advanced techniques which deal with cases where conventional asymptotics fails. This course will include asymptotics of boundary layers, WKB and multiscale methods (depending on the background of the class) and modern techniques developed to model dendritic growth (such as snowflakes), fluid flow (existence of solitary waves), and the onset of chaos. The only background needed is the basic theory of differential equations and complex analysis. Handouts Information sheet (postscript). Information sheet (PDF). Projects and Essays Mathematical Immunology Project/Essay After almost three decades of knowledge about HIV/AIDS, it was only recently discovered that a cellular automata model is capable of replicating the long three-phase cycle in clinical data on T-cell populations. However, agreement with reality is still lacking. Asymptotics of cellular automata appear to be crucial to understanding this cycle. However, currently no such theory is known. This project can go in many directions. One direction is to consider different cellular-automata models for disease transmission. A second direction is to develop mathematical methods to work out limiting behaviours of cellular automata. Nonlinear Integrable Difference Equations Project/Essay The field of integrable difference equations is only about 10 years old, but has already caused great interest amongst physicists (in the theory of random matrices, string theory, or quantum gravity) and mathematicians (in the theory of orthogonal polynomials and soliton theory). For each integrable differential equation there are, in principle, an infinite number of discrete versions. An essay in this area would provide a critical survey of the many known difference versions of the classical Painlevé equations, comparisons between them, and analyse differing evidence for their integrability. Project topics would include the derivation of new evidence for integrability. The field is so new that many achievable calculations remain to be done: including derivations of exact solutions and transformations for the discrete Painlevé equations. Exponential Asymptotics Project Near an irregular singular point of a differential equation, the solutions usually have divergent series expansions. Although these can be "summed" in some way to make sense as approximations to the solutions, they do not provide a unique way of identifying a solution. There is a hidden free parameter which has an effect like the butterfly in chaos theory. This problem has been well studied for many classes of nonlinear ODEs but almost nothing is known for PDEs and not much more is known for difference equations. This project would include studies of a model PDE, like the famous Korteweg-de Vries equation near infinity, or a difference equation like the string equation that arises in 2D quantum gravity. Home Page Postgraduate Students Recent Research Papers Last modified: 11 April 2006 by N.Joshi