Main research interests are in lattice dynamical systems. A lattice dynamical system is a regular lattice (such as a planar square lattice) where each lattice point has a state which evolves over time, and whose evolution is also affected by nearby lattice points. An example from physics is the Ising model of ferromagnetism, and an example from mathematics is Conway's Game of Life, an important example of a cellular automaton.

These pictures show the successive evolution of an Allen-Cahn type equation at T=8, T=16, T=50 and T=200. Note the rapid formation of `grain boundaries' and then their slow evolution and coarsening.

Lattice systems have been used to model many physical phenomena, particularly in materials science but also in problems of pattern recognition (using neural networks) and biological problems, where people have normally been interested in coupled oscillators.

What makes lattices interesting is that they have an intrinsic symmetry (i.e. the reflections, rotations and translations which leave the lattice looking the same) and architecture (i.e. the pattern of couplings). To get interesting dynamics we often need only nearest-neighbour (NN) coupling (two cells are coupled if and only if they are next to each other) but lattices with more extensive coupling can often give rise to more interesting behaviour.

An example of a square lattice with nearest neighbour (NN) and next-nearest neighbour (NNN) coupling.

For the present I am interested in systems which are continuous in time (i.e. they are not cellular automata, which have a discrete time evolution) and whose state at each cell is given by a real number. Even this special case has lots of interesting dynamics (and provides potential for lots of pretty pictures!)

Some possible behaviours in lattice systems. On the left we see nucleation and curvature-dependent effects, on the right a striped pattern appears.

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