Here's a little more about my research. For details, peruse my
papers or take a trip to:
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My work centers mainly around singularities: how we use asymptotics and geometry to study behaviour around them, and how they create discontinuities in dynamics. The research is with a view to applications across the physical and life sciences, often on very specific hands-on applications. But when it comes to singularities, I'm not that fussy...
Singularities...
Singularities form the bones on which the flesh of the world are laid.
Although much of the world satisfies smooth, often simple, rules (laid down in the bedrock of mathematics by Newton, Cauchy, Hamilton, Gauss, and Poincare et al), the really interesting structures are born from the outer fringes of these theories -- sonic booms, rainbows, ripples on ponds, wave-particle or wave-ray dualities. On these fringes the theories break down. In their wake, beautiful geometry reveals intricate physical phenomena, such as swirling Mach cones in helicopter noise, phantom traffic jams, the speed-of-light caustic in a high energy synchrotron, cusps, whiskers and complex rays in conical diffraction, ... and so many other things, and more that remain to be explored.
I'm still dipping my feet into these deep and sparkling waters; for someone who has spent a career diving deep beneath the surface visit asymptotico (look out particularly for a recent and scintillating history in "Taming the Tails").
Natura non facit saltus...
Our mathematical intuition seeks continuous laws of nature, but discontinuities are everywhere, in friction, switching, and impact, in neuron spiking and in collapsing quantum mechanical wave functions. Discontinuties introduce sudden irreversible changes, and worse, it seems that their singularities can produce explosions of possible futures from a single initial state. Somewhere in the borderland between smooth-but-fast change, and truly discontinuous change, nondeterminism rises from the shadows. Somewhere between singularly pertubed systems and Fillipov's "differential equations with discontinuous righthand sides", we are looking into the asymptotics and uncertainties associated with the sudden jumps found throughout nature.
and just a dab of Philosophy, should we meet in a public house somewhere...
Werner Heisenberg made the point that understanding stability in chemistry -- the stability of atoms and molecules in particular -- necessitated the invention of quantum mechanics. He went on to suggest that the stability of complex biological processes seems to require an understanding beyond current physics or chemistry. Half a century on, chaos and catastrophes contribute something to our description of living systems, but is something more radical on the horizon? Discuss...
I have been doing what I guess you won’t let me do when we are married, sitting up til 3 o’clock in the morning fighting against a hard mathematical difficulty.
Sir George Gabriel Stokes, 1857, in a letter to his young lady (Stokes 1907)
