Combinatorial approaches to problems in lattice statistical mechanics

在列举组合学和其他领域，如统计物理、计算机科学和理论化学的交界处发现的许多组合对象都是晶格动物。晶格动物出现在聚合物和囊泡模型、森林火灾、DNA变性等现象的渗流模型以及磁铁的Ising和Potts模型中。也许最著名和最通用的例子是自回避行走，它是由Orr在50多年前引入的，作为长链聚合物的几何性质的模型。我研究的总体目标是开发和应用组合工具来寻找和探索晶格模型的解决方案，特别是基于自我回避行走的聚合物的晶格模型。这些研究将有助于我们更好地了解聚合物的物理性质。我还将使用这些技术来解决模式避免排列中的未决问题。近年来，由于它们与计算机科学和代数结构研究中的问题相联系，这些对象引起了极大的关注。

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.