Algebraic combinatorics of graphs and matroids

电网络理论提供了一个自然而重要的起点，从这里开始研究数学物理、复杂分析和图论中的计数问题之间的联系。这一理论起源于19世纪--有凯利、基尔霍夫、麦克斯韦和瑞利等先驱--但近几十年来，对这一主题的现代视角带来了深刻的新见解。19世纪的材料现在已经被很好地理解了，目前统一的主题是发现这些旧的结果可以在多大程度上扩展到更一般的情况，或更准确的陈述。现代视角始于20世纪50年代的波茨模型，20世纪70年代海尔曼和利布定义的二聚体模型，以及20世纪70年代弗图恩、卡斯特林和吉尼布雷定义的随机星团模型。这些都是物理系统的数学抽象，如粘着、结晶或磁化，而电网络理论则作为一个简单的特例恢复。最令人感兴趣的是相变的问题--随着温度、压力或其他参数的变化，系统的物理行为是否会转变为本质上不同的状态？这类似于水的冻结或沸腾，但目标是在微观尺度上定量地理解这些现象。相关不平等是相变分析中的关键因素--这些相变衡量某些事件可能同时发生或相互干扰的程度。本文的主要研究方向是建立随机聚类模型的负相关不等式，补充Fortuin、Kasteleyn和Ginibre证明的正相关不等式。除了它们的物理解释，这些不等式在组合计数和概率论中也有应用。证明这些不等式所涉及的技术范围从经典的复分析和线性代数到最近在图和拟阵的组合学中的结果。在这些问题上的进展将促进我们对各种物理现象的数学模型的性质的理解。

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.