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.

Our department has made major advances in each of the following areas. We've developed a theoretical framework to describe the induced-charge mechanism for nonlinear electro-osmotic flow. Our 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, we have also been remarkably successful in unraveling some of the curious dynamics of galaxies.

Department Members in This Field

Faculty

Instructors & Postdocs

  • Peter Baddoo Fluid Dynamics, Complex Analysis, Machine Learning
  • Keaton Burns PDEs, Spectral Methods, Fluid Dynamics
  • Pui Tung Choi Applied and Computational Geometry, Metamaterials, Quantitative Biology, Medical Imaging
  • Nicholas Derr Active matter, scientific computing, continuum mechanics, poroelasticity, fluid-solid interaction
  • Valeri Frumkin Interfacial phenomena, fluidic shaping, hydrodynamic quantum analogs
  • Ousmane Kodio
  • David Kouskoulas Pilot Wave Theory, Water Waves, Quantum Mechanics, Fluid Mechanics, Acoustics
  • Bauyrzhan Primkulov Hydrodynamic Quantum Analogs, Interfacial Fluid Dynamics

Graduate Students*

  • Rodrigo Arrieta Candia Numerical methods for PDEs, Numerical Analysis, Scientific Computing, Computational Electromagnetism
  • Joel Been Nonlinear Waves, Inverse Scattering Transform, Fractional Calculus
  • Mo Chen Optimization, Scientific Computing
  • Davis Evans Hydrodynamic Quantum Analogs
  • Ivan Motorin Cluster Algebras, Resolution of Singularities, Representation Theory, Integrable Systems
  • George Stepaniants Statistical Learning of PDEs, Continuous Neural Networks

*Only a partial list of graduate students