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

Perk教授、Au-Yang教授和Scott教授将研究统计力学中完全可溶的二维模型。特别是，他们将研究手性波茨模型中杨-巴克斯特方程的高格解。此外，他们将研究不规则晶格上z不变非齐次Ising模型的自旋-自旋相关函数。手性波茨模型的确切结果将通过斯科特和皮尔斯的模型应用于生物膜。这项工作有望增加对发生在生物膜中的“波纹”相的新的深入理解。本项目是一个涉及数学物理、凝聚态理论和生物物理学的跨学科项目。数学和统计力学的相互作用可能导致纯数学领域的新发展，如结理论和函数理论。在凝聚态物质中精确可溶的二维模型的研究可以照亮生物膜的主题，这是无法直接计算的领域。

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.