I have contributed to several areas of low energy physics. The emphasis is on achieving a thorough understanding of physically realistic model problems using analytical methods, numerical investigations and simple experiments. I work on topics where there are fundamental points of principle which remain to be understood and have identified significant problems which had not previously been considered.

Areas where I made sustained contributions are listed below, in roughly chronological order. The numbers refer to my list of publications, and the contribution of my collaborators is acknowledged by listing the authors there. Publication numbers [2,34,47,50,53,77] do not fit into the categories below.

**Magnetic field effects in solids**

My Ph.D. thesis concerns Bloch electrons in a magnetic field. In
numerical experiments Hofstadter found that the spectrum is a non
self-similar fractal set; its explanation was a long standing problem.
I showed how the spectrum can be understood using a chaotic
renormalisation-group (RG) transformation. The fractal spectrum is
associated with unstable fixed points forced by a fourfold symmetry of
the lattice. Initially I approximated the RG transformation using WKB
methods [1]. Later I developed an exact form [10], which has connections
with Chern numbers describing the quantised Hall effect. This led to new
definitions of magnetic Wannier functions [32,46] and new representations of metaplectic transformations [46,51]. For lattices with threefold symmetry, the spectrum has an even richer fractal
structure, explained in [15]. An early application of the 'Berry phase' was
discovered in the WKB theory of this problem [3]. With Y. Last, I found that the
spectrum satisfies remarkable sum rules [24]. Other works in this area are
[4,7,31,38,39,42].

**Semiclassical methods**

I showed that matrix elements obey sum rules related to periodic classical orbits, analogous to Gutzwiller's celebrated relation between periodic orbits and the density of states [8,11,55]. This is relevant to periodic orbit effects in electrical conductance of mesoscopic systems. Analysis of the contributions for long orbits provides insight into connections between semiclassical results and random matrix theory [11].

I also analysed quasi-degeneracies where multiplets are associated with phase space tori which
transform into each other under a symmetry operation. The energies are split due
to a tunnelling effect [5,6,9,37].

**Energy diffusion theory of dissipation**

Most discussions of the response of macroscopic systems are based upon linear
response theory. This is problematic, because the 'Kubo formula' is based
upon quantum mechanical perturbation theory, which is questionable when the
perturbation is larger than the separation of energy levels. In applications to
macroscopic bodies, this is a serious concern. It was addressed by developing a
new approach, the 'energy diffusion theory' of dissipation, which is both
simpler and more general than the Kubo formula [17,16]. Energy diffusion theory agrees with the Kubo formula in many regimes, but in
some cases it gives dramatically different results [33,49,57,58]. I discovered a
regime with enhancement of dissipation due to Landau-Zener transitions [12,13],
and other regimes with a suppression of dissipation due to Anderson localisation
of Floquet operator eigenstates [18]. More recently, I discovered (with
B. Mehlig and D. Cohen) a dynamical behaviour
termed 'semilinear response' [66].

**Universal dynamics of complex systems**

The spectra of complex systems have
universal statistical properties, exemplified by random matrix models. I
extended this idea to universality of dynamical properties [22,33,49], was the
first to calculate statistics of the parametric dependence of energy levels
[14,23,26,27]. These were used to derive results on non-adiabatic transitions
[12], and the statistics of Chern integers (topological invariants describing
quantised Hall conductance) [35,40]. This led to general results on
the 'screening' of 'charged' singularities [48,60]. Other work on random
matrices has included theories for the parametric dependence of wavefunctions
[36], spectral correlations [54] and properties of banded random matrices [19,20,33].

**Other topics in condensed matter theory**

I explained a surprising universality in exciton spectra of heterostructures
(noted by Yang Fang), using statistical topographic ideas [25,28,43]. I have
also discovered and explained universality of an exponent describing anomalous phtoconductive response of semiconductors [87].

I made a detailed analysis of the absorption of electromagnetic radiation by small metal particles, showing how semiclassical ideas can be adapted to deal with screening of the electric field in a satisfactory way [29,30,41,44,45,52,56].

Other work on solid state physics includes a novel mechanism for hopping conduction termed 'adiabatic transport' [21], a novel 'stick-breaking' model for Mott's variable-range hopping process [75,69,73] and anomalous diffusion in solids [91].**Particles in turbulent flows**

With Bernhard Mehlig I have made extensive investigations on the motion of
particles in turbulent flows. A new method for determining the particle Lyapunov
exponents was developed [61,62,63]. We obtained an exact solution of a
'path
coalecsence transition' [59,82,90]. By developing the Lyapunov exponents as
power series and applying Borel summation, we have shown how the fractal
dimension of particle clustering depends upon the 'Stokes number', a
dimensionless measure of particle inertia [64,71,82,83,86]. We identified new
dynamical regimes, including a generalised Ornstein-Uhlenbeck process which is
analysed using novel annihilation/creation operators [65,67].

I
have also considered advection of rod-like particles in turbulence,
showing that their textures exhibit apparent singularities similar to
those of fingerprints [79,81,85,95]. With Alain Pumir I studied tumbling
in turbulent flows [89], including a surprising `irrational
quantisation' phenomenon [88].

This work also led to the definition of a new type of dimension, the spectal dimension [94].

**Planet formation and gravitational collapse**

The standard model of planet formation hypothesises that planets form by the aggregation of dust particles in a circumstellar accretion disc. Using estimates of relative velocities of particles in turbulent flows [70,78], we argued that the collision are too energetic to allow dust grains to aggregate [74]. We proposed an alternative hypothesis for planet formation, 'concurrent collapse' [91], where planets form by gravitational collapse at the same time as their star. The planets may subsequently become entrained to the accretion disc with circular orbits, and the lighter elements ablated away to leave a rocky core. This model explains way many extra-solar planets have highly eccentric orbits.

This investigation also led to new theory of gravitational collapse which
describes the role of shocks in fragmentation of a molecular cloud [84].

Rainfall

With
B. Mehlig I introduced the concept of `caustics' into studies of
inertial particles [62], and showed that this is essential to
understanding the collision rate of inertial particles. Our results
help to explain the rapid onset of rainfall from cumulus clouds [68]
(see also [72,76]).