A source of delight is uncovering down-to-earth or dramatic and sometimes beautiful [352, 363] examples of abstract mathematical ideas: the arcane in the mundane [354, 374]. Examples are:

- mathematical singularities in rainbows [213, 266] and the patterns on the bottom of swimming-pools [56];
- a laser pointer shone through irregular bathroom-window glass, illustrating abstract aspects of wave interference;
- optics with transparent overhead-projector plastic sheets, illustrating polarization singularities, matrix degeneracies, and quantum measurement [303, 361], and Anderson localization [281];
- a levitating spinning-top, illustrating adiabatic stability and geometric phases [271, 285];
- twists and turns with a belt, illustrating the behaviour of identical particles in quantum mechanics [286], responsible for the impenetrability of matter, lasers, superconductivity...
- Oriental magic mirrors [383], directly displaying the Laplacian.
- Tsunamis, which are caustics in spacetime [376], and, when focused, spacetime caustics on a cusped caustic [B]

1. |
Quantum mechanics, chaos and the primes (with Jon Keating). This concerns the relation between the Riemann zeros - harmonies in the music of the primes - and energy levels of classically chaotic quantum systems [154, 163, 227, 307, 339]. We study the statistics of the zeros [175], the asymptotics of the Riemann-Siegel formula for calculating them [233, 265], and efforts to find the operator whose eigenvalues are the imaginary parts of the zeros [306]. |

2. |
Quantum chaology for systems with mixed chaology (with Jon Keating and Henning Schomerus). Bifurcations of closed orbits cause the moments of the fluctuations of the level counting function (spectral staircase) to diverge semiclassically [294] according to power laws, whose 'twinkling exponents' depend on a competition between bifurcations [320], related to the competition between catastrophes to dominate the fluctuations of twinkling starlight [58, 114, 318]. |

3. |
Spin-statistics connection, e.g. Pauli principle (with Jonathan Robbins). Our nonrelativistic theory [286, 322] incorporates the indistinguishability of identical spinning particles is incorporated into quantum geometry in a new way. The Pauli sign is an unusual kind of geometric phase, though there is an analogue [253] for single spins. There are many physical and mathematical ramifications [319]. |

4. |
Singularities of bright light (caustics). With Christopher Howls, I have written an account of the mathematical properties of diffraction catastrophes, a new class of integrals [49, 86] with many applications to physics, including spectacular patterns of wave interference associated with focusing [79, 89, 105, 106, 256, 270]. This is part of the Digital Library of Mathematical Functions project (of which I am also a member of the editorial board), which we hope will go online on 2005. See [326] andhttp://dlmf.nist.gov/ Conical diffraction is a type of extreme singularity, now elucidated in detail after nearly two centuries [360, 381, 386, 387, 392, A]. Regarding tsunamis as caustics in spacetime facilitates an accurate calculation of the wave profile [376], and their focusing onto a cusped caustic by a submerged island is a singularity on a singularity [B]. Looking at caustics and the associated coalescing images [398] involves Husimi functions and complexified diffraction catastrophes. |

5. |
Singularities of faint light ('Optical vorticulture'). In scalar waves, these are phase singularities - that is wavefront dislocations [34, 312] or optical vortices [296] - zeros of the field; in vector waves, they are lines of purely circular or purely linear polarization, related to singularities in local expectation values of photon spin. In white light, phase singularities are decorated with universal coloured interference patterns [346, 347]. A helicoidal wavefront, whose phase pitch can be fractional, evolves into an elaborate pattern of singularities [359]. I am studying the general properties of these singularities [362, 365, 368, 395], and, with Mark Dennis, their statistics in random waves (e.g. black-body radiation) [321, 324, 340], and their knottedness [328, 332, 333,].
and their interactions [394]. In crystal optics, singularities appear in the space of directions, along wth a new type of degeneracy singularity; this viewpoint leads to clarification and simplification of this old subject, especially for crystals that are absorbing and optically active (chiral) as well as anisotropic [355] and, bianosotropic [379], and the polarization pattern of the blue sky [373]. The phase singularities of classical optics are windows, through which can be glimpsed the faint glimmering of the quantum vacuum [364]. |

6. |
Asymptotics and relations between theories.
Asymptotics and relations between theories. Less general theories in physics are limits of more general ones, but the limits are usually singular [260, 341]. An example is classicalization through decoherence, where tiny uncontrolled external influences suppress the quantum suppression of classical chaos [337]. Refined studies of these singular limits must involve divergent series, Stokes phenomenon [181, 190, 201], and (work with Christopher Howls) resurgence and hyperasymptotics [208, 223, 244, 249, 261], and band-limited functions can oscillate arbitrarily faster than their fastest Fourier components [252, 262, 388]. Surprisingly, the exponentially small contributions whose appearance and disappearance is governed by the Stokes phenomenon [181] can sometimes dominate [370]. Another universal asymptotic phenomenon is oscillations generated by infinitely repeated differentiation [377]. Asymptotics is a unifying idea: the same mathematics that governs rainbows also describes tsunamis [376]. These studies are part of humanity's long struggle to get to grips with infinity [241]. |

7. |
Nonhermitian operators. When there is absorption or loss, physical operators can be strongly nonhermitian, and much of the resulting new physics is associated with energy-level degeneracies [372], as in near-resonant atom optics [293, 295], and unstable lasers [332, 334, 350]. Crystal optics singularities [355, 379] (see section 5 above) exhibit a variety of nonhermitian phenomena. When the nonhermitian operators have antiunitary symmetry (e.g. PT), spectra can be real [325, 345]. |

8. |
Extreme coherence.
The addition of many discrete waves with quadratically-varying phases, as in the near field of diffraction gratings or in time-dependent quantum mechanics of periodic waves, striking interference phenomena occur, dominated by the Gauss sums of number theory. Such hypercoherent waves generate fantastic spirals in their complex planes
[171, 179],
and elaborate carpets in space (Talbot effect) or spacetime (quantum revivals)
) [274, 275, 283, 304, 329]. |