On this year's to do list...
- Get Hidden Dynamics into the bookshops. (Book written and on the way).
- Finish paper on analytic curvature: complex variables beyond Cauchy-Riemann.
- Kickstart the next big events in Nonsmoothland. (COST? INI? Conference?)
- Get 10km time back down to 40 minutes.
- Persuade my new son to stop sleeping sideways with his feet up against the side of his cot.
What I'm working on right now...
- illusions of noise, and
- What do they say about determinacy in the real world?
- How do we recognise and apply them across the sciences?
Our studies of discontinuity in dynamics have revealed two big surprises:
Now we need to put these into action:
Who am I?
I'm a researcher who studies mathematics with a view to understanding the world around us. The simple and surprising fact is that we still possess only the beginnings of the mathematics required to understand the world around us. So motivated by anything interesting in science, nature, and engineering, I poke at the geometry and singularities underneath, looking for the kind of deep insights that only mathematics can give.
Seemingly complex things often have a simple cause. There is a way of looking at the world that searches out its patterns, studies how they change through dynamics, how they are connected through networks, and how they interact through forces. Applied mathematics encodes all this into equations, always aware that these are but approximations, toy models, of reality. But it is not their distance from our reality that limits the power of our models, rather it is their distance from our intuition.
We expect a good model to be smooth and deterministic, to behave broadly similarly in all instances, so that we can predict how it changes. But the world is less smooth and less deterministic than the mathematical ideals we have clung to for centuries. Fortunately, mathematics has no such prejudices, but lies patiently waiting, as it has done many times, for our slow attitudes to change. It turns out that discontinuity and non-determinism are already embedded in the very elemental fibre of mathematics. And one way they appear is via singularities.
You know what singularities are. Sure, you've heard of black holes. But they're also the cause of rainbows and sonic booms, rattling wheels and squealing brakes, they generate your very thoughts, they are phantom traffic jams with no apparent cause, and the reason why teasing a pleasant note from a violin takes particular skill.
Singularities provide structure to a world that, without them, would be too polished and too perfect to be any fun. They are the bones on which the flesh of the world is laid. For mathematics they are anaethema, for applied mathematics they are the doorway to reality.
What do I do?I while away the hours with pad and pen or, when possible, with a blackboard and chalk, in my comfy home in Engineering Mathematics at Bristol or on occasional forrays around the world, collecting interesting teas and intriguing ideas wherever I go. My work has brought me into contact with interesting minds spanning three centuries and seven continents. I teach a little because it's fun. I lecture because I want to hear people say "I think of that a different way...!"
And why?Francis Bacon said "mathematics should only give limits to natural philosophy, not generate or beget it". I hope that isn't true. While Bacon was laying cornerstones of empirical science, I hope he would eventually have been swayed by Dirac's vision of a mathematics that can breach the boundaries of natural philosophy, of a tool more skillful than it's smith's hands, always one step ahead of our imagination. For some years now I've been trying to figure out whether nonsmooth dynamics limits our understanding of practical physics, or throws it wide open, or merely paves a road to obscurity. The results are now ready and have taken us by surprise.
And if you can handle a little jargon...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. Like singularities (and often caused by them), 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.