Below is a synopsis of my research interests, with a more detailed list of possible research topics and directions for potential students and collaborators on the topics page. My publications are listed and many talks are available for download on my publications page. Please visit the projects page if you are interested in seeing visualisations of my work.
My main field of research is statistical mechanics, which is the study of systems with many, many constituents, with the goal of understanding how these constituents can cooperate to bring about global changes of state, otherwise known as phase transitions. Typical examples of phase transitions are liquid water freezing, or boiling, or a magnet losing its magnetic field when it's heated.
In particular, I study various mathematical models of statistical mechanical systems, and exploit the fundamental property of universality. Universality guarantees that no matter how simple or idealised a model is, provided it is in the same “universality class” the model will have exactly the same fundamental properties as real, physical systems.
One example of universality, and in fact the one I'm most familiar with, is the way self-avoiding walks exactly model certain aspects of polymers (long molecules) in a solvent. Self-avoiding walks are an extremely simple model to define: just take a grid in two or three dimensions, and start off at one of the sites. Then walk to neighbouring sites for N steps, with the the rule that you cannot return to a previously visited site. It turns out that these self-avoiding walks exactly capture important features of polymers in a good solvent, such as the relationship between the size of a polymer (what size ball it can fit inside) and the length of the polymer.
For my PhD I worked with Barry McCoy at Stony Brook and made some improvements in the numerical and analytical calculation of virial coefficients for hard spheres. Hard spheres are an excellent model of colloids and are historically important for a host of reasons, including the fact that when compressed the hard sphere system changes state from a fluid to a solid. However, much remains to be done in this direction! Progress depends on implementing an efficient algorithm for the calculation of a graph theoretic property that is related to the chromatic polynomial.
More recently, my primary research focus has been to build upon the intuition I have gained into the efficient Monte Carlo simulation of self-avoiding walks via the pivot algorithm. Significant progress is possible for a wide range of applications, both in the field of statistical mechanics (percolation, Hamiltonian paths, hard spheres) and more broadly (exotic option pricing), for which either simple sampling or Markov chain Monte Carlo is the state of the art. (Jason Whyte has written a lucid explanation of the basic idea at the SAW feature page.)
I'm interested in many other topics (it may be shorter to list things I find uninteresting!), but in particular I have research interests in developing efficient enumeration algorithms, physical combinatorics, climate change (especially policy around methane and carbon dioxide, so-called multi-gas modelling), modelling more broadly, polymer physics, and mathematical visualisation.