Algebraic combinatorics of graphs and matroids

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The theory of electrical networks provides a natural and significant starting point from which to begin investigating the connections among mathematical physics, complex analysis, and enumeration problems in graph theory. The origins of this theory are in the 19th century -- with pioneers like Cayley, Kirchhoff, Maxwell, and Rayleigh -- but a modern perspective on the subject has led to deep new insights in recent decades. The 19th century material is now very well understood, and the current unifying theme is to discover the degree to which these old results can be extended to more general situations, or to more precise statements. The modern perspective begins with the Potts model in the 1950s, the dimer model defined by Heilmann and Lieb in the 1970s, and the random cluster model defined by Fortuin, Kasteleyn, and Ginibre in the 1970s. These are mathematical abstractions of physical systems such as adhesion, crystallization, or magnetization, and the theory of electrical networks is recovered as a simple special case. Of central interest is the question of phase transitions -- as the temperature or pressure or other parameters change, does the physical behaviour of the system undergo a shift into a qualitatively different state? This is analogous to the freezing or boiling of water, but the goal is to understand the phenomena quantitatively at a microscopic scale. Correlation inequalities are key ingredients in the analysis of phase transitions -- these measure the extent to which certain events are likely to occur together, or to interfere with one another. The main direction of my research is to establish negative correlation inequalities for the random cluster model, complementing positive correlation inequalities proved by Fortuin, Kasteleyn, and Ginibre. Along with their physical interpretation, such inequalities also have applications in combinatorial enumeration and probability theory. The techniques involved in the proof of such inequalities range from classical complex analysis and linear algebra to very recent results in the combinatorics of graphs and matroids. Progress on these questions will advance our understanding of the properties of mathematical models of various physical phenomena.