Combinatorial approaches to problems in lattice statistical mechanics

在计数组合学和其他领域（如统计物理学、计算机科学和理论化学）之间的界面上发现的许多组合对象都是格子动物。晶格动物出现在聚合物和囊泡的模型中，出现在森林火灾、DNA变性等现象的渗透模型中，出现在伊辛和波茨的磁铁模型中。 也许最有名的和通用的例子是自回避步行，这是由奥尔介绍了超过50年前作为一个模型的几何性质的长链polymer.The我的研究的总体目标是开发和应用组合工具来寻找和探索解决方案的晶格模型，特别是，晶格模型的聚合物的基础上的自回避步行。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避免模式的排列中的未决问题。近年来，由于它们与计算机科学和代数结构研究中的问题的联系，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.