Research Interests

My general interest lies in lattice path problems in combinatorics and statistical mechanics. They are often problems that are simple to state but require the most sophisticated techniques to solve. They are used in a variety of modelling situations, including physical chemistry applications of polymers in solution, and to computer science applications related to computer language construction.
     As an example, consider a set of paths of a certain length on a directed square lattice starting and ending at some chosen set of points. The paths are not allowed to have any vertices in common. How many different configurations are there? For certain special starting and ending positions the number of such configurations can be written as a product form. I am trying to find combinatorial proofs of these product forms. If the paths can have vertices in common then they are called osculating lattice paths and are related to the six-vertex model of magnetism and the combinatorial problem of enumerating alternating sign matrices. I am currently characterising the combinatorics of osculating lattice paths.
      I collaborate with Professor J. Essam of the University of London on these topics as well as local colleagues Prof. Tony Guttmann and Dr Aleks Owczarek. Andrew Oppenheim (Masters student) is also working with me in this area.