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Research interests

  • Geometrical Asymptotics. Although the bulk of the world satisfies nice simple rules laid down from Descartes, to Newton, Hamilton, through to Dirac, 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, but in their wake, beautiful geometry reveals intricate physical phenomena, such as swirling Mach cones in helicopter noise, the speed-of-light caustic in a high energy synchrotron, cusps, whiskers and complex rays in conical diffraction; ...

  • Grazing Catastrophes. The theory of dynamical systems with discontinuities, such as friction, switching, or impact, is young and growing. At the heart of most theories is the assymption of smoothness, but from the squeal of brakes to the rattle of a fan blade, it is clear that in the real world things rarely run smoothly. These events are discontinuous, they involve sudden changes and loss of reversibility, and occur whenever two systems come into contact. Mathematics was not built to handle discontinuities, so we treat the interaction ad hoc, we look at the "before" and "after" in isolation, then stitch things back together later. But perhaps nature is not so malicious. Perhaps not everything is lost when a system jumps. By studying orbits that "graze" discontinuities - that almost jump but not quite - we hope to get to the heart of discontinuity.

  • From nonsmooth to slow-fast - Discontinuity Asymptotics. What does the borderland between a system with a discontinuity, and a system with a sudden (slow-fast) change, look like? Experimental laws of the second type are incredibly complicated. Theoretical equations of the first type are too idealistically simple. Somewhere between them are the bifurcations and the catastrophes that define real world events everywhere from engineering to physiology, where contact between systems takes place much more quickly than changes in the systems themselves, but hugely affects their dynamics. I am interested in a nonsingular understanding of singular perturbation theory: a geometrical relationship between nonsmooth and slow-fast systems.


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