What I'm reading right now...
- Barbour's Discovery of Dynamics;
- new papers on old problems: if friction and impact are "classical" mechanics, why are they still so mysterious?
- Bender and Orszag's Methods
- the cover of Meat Loaf's Hell in a Handbasket.
What I'm working on right now...
The nonsmooth world finally cracked open, and out fell:
- dummy dynamics,
- switching hierarchies, and
- determinacy breaking events.
Haven't heard of them? Don't worry, no-one has, yet.
Who am I?
I'm a researcher who studies maths with a view to any application in science and engineering. Basically, surprisingly, we don't yet have the mathematics we need to even try to understand the world around us. So motivated by anything interesting in science, nature, and engineering (actually almost anything except computer technology), I try to figure out what mathematics is bubbling underneath to make it work.
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 control your very thoughts, they are 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. 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 laid the cornerstones of empirical science, I hope he would eventually have been swayed by Dirac's vision of a mathematics that can burst the boundaries of natural philosophy apart, of a tool more skillful than it's smith's hands, always one step ahead of our imagination. Right now I'm 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.
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.