Mathematical Sciences: Exact Analysis of Lattice-Statistics Models and Applications to Biomembranes

帕克、欧阳和斯科特教授将研究统计力学中精确可溶的二维模型。特别是，他们将研究手征Potts模型中杨-巴克斯特方程的高亏格解。此外，他们还将研究不规则晶格上Z不变非均匀伊辛模型的自旋-自旋关联函数。手性Potts模型中的精确结果将通过Scott和Pearce的模型应用于生物膜。这项工作有望增加对生物膜中“涟漪”阶段的新的深入理解。这个项目是一个涉及数学物理、凝聚态理论和生物物理学的交叉学科。数学和统计力学的相互作用可能导致纯数学在纽结理论和theta函数理论等领域的新发展。对凝聚态物质中精确可溶的二维模型的研究，可以阐明计算无法直接进入的领域中的生物膜这一主题。

Professors Perk, Au-Yang, and Scott will study exactly soluble two dimensional models in statistical mechanics. In particular, they will investigate the higher genus solutions of the Yang-Baxter equations in the chiral Potts model. Furthermore, they will study the spin-spin correlation functions of the Z-invariant inhomogeneous Ising model on an irregular lattice. The exact results in the chiral Potts model will be applied to biomembranes through the model of Scott and Pearce. This work promises to add new in-depth understanding of the "ripple" phase which occurs in biomembranes. This project is an interdisciplinary one involving mathematical physics, condensed matter theory, and biophysics. The interplay of mathematics and statistical mechanics could lead to new developments in pure mathematics in areas such as knot theory and the theory of theta functions. The study of exactly soluble two dimensional models in condensed matter could illuminate the subject of biomembranes in areas which are not directly accessible to computations.