Research
My research centres on using the tools of Analysis to understand and improve models arising in Materials Science.
A substantial thread of my research focuses on questions relating to the mathematics of solid materials, and principally on crystals. Crystals are made up of atoms laid out in a regular repeating pattern, and include most metals at room temperature and pressure.
I am particularly interested in modelling defects and microstructure of these materials, i.e. the behaviour of small-scale phenomena which affect material properties of crystals on much larger scales. Defects are known to play an important role in the behaviour of plastic (or irreversible) deformation of crystals, and ultimately govern their failure. Deriving and improving predictive models for the evolution of microstructure therefore has important potential consequences for engineering applications.
For further detail on this research theme and a list of my collaborators and publications in this area, see here.
A more recent theme of my research I have begun working on is the theory of coarse-graining for dynamical systems, particular those used in molecular simulation. Coarse-graining techniques attempt to reduce complex mathematical models to simpler ones for practical computation. One approach to performing this reduction is based on the Mori-Zwanzig formalism, which uses projection operators to inspire data-driven numerical approximation techniques.
For further detail on this research theme and a list of my collaborators and publications in this area, see here.
Analytical tools and techniques
I have used a wide variety of analytical tools in my research to prove approximation results connecting different models, often at different temporal and spatial scales. These methods are drawn from across Applied Mathematics, the Calculus of Variations, PDE Analysis and Stochastic Analysis, and include: formal asymptotic expansion, Gamma-convergence, quantitative versions of the Inverse Function Theorem, Large Deviations Theory, and Stochastic Homogenization.