Research Interests

I am primarily interested in the dynamics of nonlinear PDE and particularly singularity formation. In looking at the structure of singularity formation we need sensible models, asymptotic reduction, sophisticated numerical computations and rigorous analysis. My scientific interest lies at the interface of all thses approaches. Most of blow-up problems have a self-similar structure that can be exploited both analytically (PDEs become ODEs) and numerically (adaptivity based on scale invariance). I am both constructing and using equidistribution based adaptive methods in one, two and three space dimensions. These computations provide insight and direction for both formal and rigorous analysis. Currently I am mostly working on a number of problems related to blow-up: a variational characterization for blow-up in an unstable thinfilm equation, blow-up rates in some semilinear wave equations, stability of singular radial profiles in harmonic map heat flows and reliable numerical methods for PDEs with finite-time singularities. Beyond this my new colleagues have got me interested in bifurcations in some fourth-order (non-singular) PDEs and simulation methods for molecular dynamics that couple both discrete and continuous representations. There are too many interesting problems!