Simon Malham

Accepting PhD Students

PhD projects

Non-commutative integrable systems---------------------------------------------This project concerns the integrability of non-commutative nonlinear partial differential equations. In particular, the non-commutative Kadomtsev-Petviashvili (KP) hierarchies, and their modified forms, are very much of interest. Establishing their integrability by direct linearisation would be one goal. An underlying operator algebra, the pre-Poppe algebra, provides a natural context for establishing direct linearisation for these hierarchies. These hierarchies have an incredibly rich structure and many applications, for example, in nonlinear optics, ferromagnetism and Bose-Einstein condensates. They have intimate connections to: string theory and D-branes; Jacobians of algebraic curves and theta functions; Fredholm Grassmannians; the KPZ equation for the random growth off a one-dimensional substrate; and so forth. In addition to these aspects, there are many further directions that could also be explored, for example: the log-potential form; super-symmetric extensions; establishing efficient numerical methods via the direct linearisation approach; etc. Coagulation sol-gel phenomena---------------------------------------This project concerns Smoluchowski coagulation and sol-gel models. We consider scenarios where particles of different sizes coalesce to form larger particles, including possibly a "gel" state. There are many applications including: aerosols, clouds/smog, clustering of stars/galaxies, schooling/flocking, genealogy, nanostructures on substrates such as ripening or island coarsening, blood clotting and polymer growth, for example, in biopharmaceuticals. The goal is to find analytical solutions as well as construct efficient numerical simulation methods. Determining the dual sol-gel state is an important aspect of these models. Planar tree structures play an important role as well, both at the nonlinear partial differential equation model level, and at the particle model level where, naturally, coalescent stochastic processes represent their overall evolution. Including spatial diffusivity in the model at both these levels, for example to model colloids, adds another complexity. There are many directions to explore, for example: the particle interactions can be much more complex; there is a natural Hopf algebra of planar trees that is likely useful in optimising numerical simulation; investigating the dual reverse time, branching Brownian motion, perspective; and so forth.