Combinatorial approaches to problems in lattice statistical mechanics

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Many of the combinatorial objects found at the interface between enumerative combinatorics and other fields, such as statistical physics, computer science and theoretical chemistry, are types of lattice animals. Lattice animals appear in models of polymers and vesicles, percolation models of phenomena such as forest fires, DNA denaturation and the Ising and Potts models of magnets.  Perhaps the most famous and versatile example is the self-avoiding walk, which was introduced by Orr over 50 years ago as a model of the geometric properties of long chain polymers.The general aim of my research is to develop and apply combinatorial tools to find and explore solutions to lattice models; in particular, lattice models of polymers based on the self-avoiding walk. These studies will help us better understand physical properties of polymers. I will also use these techniques to address outstanding open problems in pattern-avoiding permutations. These objects have attracted a great deal of attention in recent years due to their links with problems in computer science and in the study of algebraic structures.