Mike    Jeffrey

Welcome...

• Who am I?

  I'm a researcher who studies mathematics with a view to understanding the world around us. I like wandering the world with a pad and paper wondering about why things do what they do, and how to capture them in equations.

  But you can't capture everything. My work has led me to focus on singularities -- places where the laws governing the world, if such things exist at all, break down, fall apart, leaving rents and holes in the fabric of an otherwise calmly changing environment.

  That may sound like an odd place to start, but singularities provide the bones on which the skin of the world is laid, shaping the smoother features that lie in between. They are the geometry underpinning everything from black holes to economic upheavals, from sonic booms and rainbows to capsizing ships.
  
  By studying behaviour near singularities we can exprapolate to understand the broad swathes of the intervening patches of calm. They are important because things tend not to work the other way: studying patches of calm will rarely let you understand singularities, or anything far from your starting point at all.

  In the last decade I have largely focussed on a particular kind of singularity -- a discontinuity. A discontinuity is what happens when hard objects collide, when switches are flipped, or when decisions are made. Studying these is beginning to reveal some real surprises about how our world works.

• If you're not an academic and you've found your way here...

  Welcome! Now your first thought may be it sounds interesting but I never really got maths!. But maths is not so different to music, in that we can experience and practice both everyday without having to understand them. Just as we listen to songs, so we read the time, or our speedometers, or prices. Just as we hum along to old favourites, so we estimate whether we'll be in time for that next meeting, or have enough money for that holiday. Not many of us are thinking about dissonant intervals or chord progressions as we join in with music, just as we're not solving quadratic equations and integrals as we're rushing down the M4 or M6 estimating how many miles are left in the tank.
  
  So maths has a big role in all our lives, we just don't notice most of it washing over us. There's no harm in taking a closer look behind the scenes, in trying to see how many of those bar chords or harmonies you can master.

• 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...!"

  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. It is not the distance from our reality that limits the power of our mathematical models, rather it is their distance from our intuition . . . we are still only just learning how the world really works.

• And if you were expecting something more technical ...

  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 other singularities, discontinuities are not a disease aflicting Nature, but part of it's vital substructure. And we are still just learning how to handle them mathematically, and what their role in the world is.

  The world isn't really smooth, but its not entirely nonsmooth 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.