bristol.ac.ukWhat I'm reading right now...
- Milnor's Morse Theory;
- Kühn's forthcoming Multiple Time Scale Dynamics, keep a look out;
- Dingle's and Erdelyi's viewpoints on Asymptotic Expansions;
- the sleeve of Springsteen's Bound For Glory.
What I'm working on right now...
- two-fold singularities: noisy ones and n-dimensional ones,
- a Victorian light cast on Georgian dynamics to ask: where do discontinuities come from?
- ducks (or "canards"): pinching ducks, twisting ducks, ducks under microscopes, and kinky ducks (no animal cruelty involved, and no levitating ducks).
Who am I, what do I do, and why?
Bath perturbations...
I am currently at Bath enjoying the freedom of an EPSRC CAF Fellowship, but rumour has it I will shortly be settling down in Bristol. For the last few years I have been studying singularities, first in optics and electromagnetism, and later in dynamical systems. Singularities are the columns on which Nature builds her world, the structure behind its geometry. I use them to study how the structure of systems change (bifurcations), and to tame complex behaviour using the tools of asymptotics and singular perturbation.applications...
I study whatever little comes within the grasp of the tools at my disposal. Some exquisite 1830s crystal optics, synchrotron radiation caustics, sonic booms from helicopter blades, and phantom traffic jams -- all have the same singularities at their heart. At the moment I am troubled by discontuities in dynamical systems ("piecewise smooth ordinary differential equations" for the initiated). I am interested in models of dynamics including: weather systems, the nervous system, chemotaxis in biological cells, friction and impacts, electronic switches, and economic or social decision making. All of these are plagued by discontinuties. Actually, "plagued" is the wrong word. Like singularities, discontinuities are not a disease, but a part of Nature's vital substructure. And we are still just learning how to understand them.and discontinuities...
Perhaps you like your universe polished, so that "sudden" changes appear smooth under enough magnification. Or may be you embrace imperfection and like your universe tessallated like those quantum chaps. Whichever side gets your vote, the prevalence of sharp jumps in the world is obvious, both physically and theoretically, from Coulomb friction to spiking neurons to collapsing wave functions. From the struggle to understand system-level dynamics in a complex world, to new(ish) fundamental mathematics such as the theorems of slow manifolds, multi-scale calculus, Filippov systems and piecewise smooth flows. New fundamental concepts have emerged -- blow up, canards and mixed modes, sliding and grazing bifurcations. These and much, much, more, line the road to understanding the asymptotics of interacting systems.