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Physical Applied Mathematics

This area has two complementary goals:

1. to develop new mathematical models and methods of broad utility to science and engineering; and

2. to make fundamental advances in the mathematical and physical sciences themselves.

The department has made major advances in each of the following areas. Researches have developed a theoretical framework to describe the induced-charge mechanism for nonlinear electro-osmotic flow. Their work in biomimetics focuses on elucidating mechanisms exploited by insects and birds for fluid transport on a micro-scale. These and other activities in digital microfluidics and nanotechnology have applications in biologically inspired materials such as a unidirectional super-hydrophobic surface, and devices such as the `lab-on-a-chip' and micropumps. The theory of transport phenomena provides a variety of useful mathematical techniques, such as continuum equations for collective motion, efficient numerical methods for many-body hydrodynamic interactions, measures of chaotic mixing, and asymptotic analysis of charged double layers. Nanophotonics is the study of electromagnetic wave phenomena in media structured on the same lengthscale as the wavelength, and is an active area of study in our group, for example to allow unprecedented control over light from ultra-low-power lasers to hollow-core optical fibers. New mathematical tools may be useful here, to give rigorous theorems for optical confinement and to understand the limit where quantum and atomic-scale phenomena become significant. Granular materials provide challenging problems of collective dynamics far from equilibrium. The intermediate nature (between solid and fluid) of dense granular matter defies traditional statistical mechanics and existing continuum models from fluid dynamics and solid elasto-plasticity. Despite two centuries of research in engineering, no known general continuum model describes flow fields in multiple situations (say, in silo drainage and in shear cells), let alone diffusion or mixing of discrete particles. A fundamental challenge is to derive continuum equations from microscopic mechanisms, analogous to collisional kinetic theory of simple fluids. On a far larger scale, they have also been remarkably successful in unraveling some of the curious dynamics of galaxies.

Computational Science & Numerical Analysis

Computational science is a key area related to physical mathematics. The problems of interest in physical mathematics often require computations for their resolution. Conversely, the development of efficient computational algorithms often requires an understanding of the basic properties of the solutions to the equations to be solved numerically. For example, the development of methods for the solution of hyperbolic equations (e.g. shock capturing methods in, say, gas-dynamics) has been characterized by a very close interaction between theoretical, computational, experimental scientists, and engineers.

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