Random Knotting and Lattice Paths

纽结理论在分子生物学中起着重要的作用，因为DNA分子中的缠结可以影响细胞过程，如复制和转录。 我的大部分研究都是为了理解随机打结的现象。 我特别感兴趣的格模型，其中组合方法可以使用，我的目标是了解不同的结类型的相对发生频率，并得出结概率的界限。 现代显微操作技术意味着单个聚合物分子可以受到力的作用。 我的研究旨在构建和解决格模型，帮助理解和解释这些 测量，以及更一般地，聚合物如何响应于所施加的力，特别是当聚合物从表面被拉动或从紧凑的形式被拉动到延伸的形式时。 主要目的是研究精确可解的模型，使用组合方法，但我也使用数值方法，如蒙特卡罗技术，严格的解决方案是遥不可及的。

Knot theory has played an important role in molecular biology since entanglements in DNA molecules can affect cellular processes such as replication and transcription. Much of my research is aimed at understanding the phenomenon of random knotting. I am especially interested in lattice models where combinatorial methods can be used and my aims are to understand the relative frequency of occurrence of different knot types, and to derive bounds on the knot probability. Modern micromanipulation techniques mean that individual polymer molecules can be subjected to a force. My research is designed to construct and solve lattice models that help to understand and interpret these measurements and, more generally, how polymers respond to applied forces, especially when a polymer is pulled from a surface or pulled from a compact to an extended form. The primary aim is to study exactly solvable models, using combinatorial methods, but I also use numerical methods such as Monte Carlo techniques where rigorous solutions are beyond reach.