Mike    Jeffrey

Who am I, what do I do, and why?

• Bristol geometry...

  Ensconsed with a blackboard, an EPSRC CAF Fellowship, and a small collection of interesting teas, in the surroundings of Royal Fort Gardens and the hallways of Engineering Mathematics at Bristol, I try to make a little sense of the world from the viewpoint of singularities and asymptotics. Singularities provide structure to a world that, without them, would be too polished and too perfect to be any fun. A world of focusing in optics and electromagnetism, of shifting stability in mechanics. These worlds are classical, even antiquated. So how can it be that we still have so little theoretical grasp of a process as fundamental as interaction, of its integral phenomena like friction, or impact? Is it because they involve the coming together of different scales, different media? Because interactions take place on the frontier between theoretical ideals, on badlands characterised as singular, divergent, and inhabited by the many "non"s of complex dynamics: nonlinear, nonsmooth, nonunique, nondifferentiable, nonstandard, and nondeterministic.

• ... applications...

  I study whatever little of systems physical, biological, or social, comes within the grasp of the tools at my disposal. I've studied some exquisite 1830s crystal optics, synchrotron radiation caustics, sonic booms from helicopter blades, and phantom traffic jams, all of which 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 precisionist), and am interested in models including: weather systems, the nervous system, chemotaxis in biological cells, friction and impacts, electronic switches, and economic or social decision making. And almost anything else.

• ... and discontinuities...

  Divergent series, catastrophes, singular perturbations, whether you'd prefer to believe in a polished universe or one full of jumps, discontinuities are a fact of mathematical physics, in both theory in application. All of the applications mentioned above are plagued by discontinuties. Actually, "plagued" is the wrong word. Like singularities, discontinuities are not a disease aflicting Nature, but part of it's vital substructure. And we are still just learning how to understand them. The world isn't as smooth as it used to be, but its not perfectly unsmooth either (it's erfy, see here ). It's not always convergent, not alway integrable. From Coulomb friction to spiking neurons to collapsing wave functions. From the struggle to understand system-level dynamics in a complex world, to new geometrical concepts such as slow manifolds, multi-scale calculus, and piecewise smooth flows. Fundamentally new ideas are emerging all the time in this nascent arena of mathematics, things like blow up, canards and mixed modes, sliding and grazing, the curse of dimensionality, and the two-fold path to broken causality. These and much, much, more, line the road to understanding the asymptotics and discontinuities of interacting systems.

Visit me in Nonsmoothland, or peruse my Research Interests and Papers.