Figure of eight knot flow: There is a famous flow on the complement of a figure of eight knot but as far as I can make out no-one has ever drawn its phase portrait. Understand the construction sufficiently to work it out by hand or by computer. A good starting point is some papers of Rob Ghrist, Georgia Tech (he has a nice power point presentation on his website too).
Poincare second species orbits: These are solutions of the three body problem of celestial mechanics with one large and two small masses, in which the two small masses approximately follow independent Kepler ellipses except at near collisions with each other. For the planar circular restricted case, work out some of the possible limits of second species orbits as the second mass goes to zero and draw pictures of them in both the absolute frame and the frame rotating with the second mass (you will need a computer for the latter). Start from a paper of mine with Bolotin (Cel Mech 77 (2000) 49, also available from my website).
Robot walking: Find stable periodic orbits of a model of a bipedal robot walking down a slope (that I can give you on request), and examine their robustness to non-uniformities in the slope and to time-dependent external forces.
Simplest Anosov linkage: Tim Hunt and I showed that a triple linkage has a parameter regime with Anosov energy levels, but its configuration space has genus 3. Find a linkage whose configuration space is a surface of genus 2 and look for parameter regimes in which it has Anosov energy levels. Start from my paper with Hunt (Nonlinearity 16 (2003) 1499, also available from my website).
Fast Fourier transform on the 2-sphere: Understand the status of this (a starting point is the website of Dan Rockmore, Dartmouth) and try to improve on current algorithms, with a view to assisting computations of dynamo action in the earth's core by people like Chris Jones, Exeter.
Multiple criterion optimisation: This method (under various similar names) has become extremely popular among social scientists. Work out what it is and assess its uses.
Protein fracture: Proteins in Peter Derrick's cyclotron mass spectrometer (Chemistry) appear to fracture under collision by water molecules at places related to secondary structure. Explain the places. Try to do this by imagining the only degrees of freedom in the backbone are the torsion angles about C-Calpha and Calpha-N bonds and that fracture results from exerting excessive torque on a peptide bond. So the configuration space for an N-amino-acid backbone modulo isometries is a torus of dimension 2N-2-P (where P is the number of prolines), minus holes corresponding to steric hindrance. Imagine a segment of K amino-acids between two peptide bonds which are held in some relative position R in R^3 x S)(3) with respect to each other. Generically the space of possible configurations of the K-peptide is a (disjoint union of) 2K-P-6 dimensional manifolds, e.g. a finite set of points for K=3, P=0. Find the singularities of the map from the (2K-P)-torus to R^3 x SO(3). The idea is that if R crosses a fold then infinite torque is exerted on at least one peptide bond and so it will break.
Lagrange representation for configurations of particles: Lagrange proposed to represent configurations of N particles in 3 dimensions modulo isometries by non-negative quadratic forms of rank at most 3 in (N-1) variables, which can be thought of as neutral "charge" distributions on the N particles, the quadratic form being the resulting squared dipole vector. See if this could be useful for some purpose like NMR structure determination of proteins in solution. A starting point is a paper by Albouy and Chenciner.
Transitions of DNA under external forces: Bryant et al (Nature, 2003 or so) report on experiments on (double-stranded) DNA in which they twist and pull one end relative to the other. They interpret in terms of segments of the DNA overtwisting or denaturing. Formulate and study statistical mechanics models for such situations. Include the case (for which I am not aware of an experiment) of pulling the ends of the strands apart (leaving the other paired end free for example). Predict the temperature dependence of some time-averaged quantities, eg distance between strand ends as a function of applied force.
Photosynthetic antennae: The light-harvesting systems of most photosynthetic systems consist of rings of photoreceptive molecules. Explain the advantage this gives in terms of turning an absorption line into an absorption band. Understand the cascade process between the concentric rings.
Nonlinear conductivity by thermal nucleation of pairs of kinks: Try to explain nonlinear conductivity of charge density wave materials in terms of Frenkel-Kontorova chains, for which there can be different thresholds for sliding of a kink or antikink from that for a periodic structure. See recent book by Brown and Kivshar for background on Frenkel-Kontorova models and Hirthe and Lothe for conductivity by thermal nucleation of defects.
Bayesian function fitting: see my website list for 03/04
Alloy structure: see 03/04
Turbine blades: see 03/04