Algebraic methods for lattice models of statistical physics

物理学通过原子或分子结构来理解物质。这种描述在预测物理性质和发现新材料方面取得了压倒性的成功。它特别适合于原子和分子数量有限但不是太大的系统。只有当大量原子或分子相互作用时才会出现的现象需要其他技术。相变就是这样一种现象：众所周知，除非相互作用的物体的数量趋于无穷大，否则不可能发生相变。拟议的项目旨在理解从有限系统(例如，那些原子数量有限的系统)到它们的连续极限(从一开始就假设有无限多个相互作用的对象的理论)的过程。特别是，它想要确定有限系统的属性，这些属性揭示了通过增加对象数量而获得的无限系统的属性。该项目的主要数学工具将是代数。数学的几个章节在描述相变方面发挥了作用。分析和代数可能是核心部分。代数提供了识别物理系统对称性的工具。对称是在不改变其整体物理属性的情况下将一个系统转换为另一个系统的操作。这些对称性提供了接近物理系统并从数学上定义它们的基本方法。该项目重点研究了有限格子模型及其无限连续极限中的代数结构。这项研究将集中在微观相互作用的二维晶格模型上。这些模型被称为渗流模型、伊辛模型、XXZ自旋链、密集和稀疏回路模型等。它们为探索物理性质并严格证明它们提供了一个天然的实验室。它们有有限数量的“粒子”，它们可以在计算机上探测，它们被认为是(对数)共形场理论(一组著名的、研究得很好的连续介质模型)。最重要的是，它们依赖于一种代数描述，这自然有助于研究大量粒子的极限。以前在这些方向上的工作，其他人和我的工作，都对物理和数学做出了贡献。例如，在物理学中，认识有限系统的新兴性质是有用的，即使只有部分实现。在数学中，对物理系统的代数结构的研究提出了许多新的发展途径或看待现有结果的新方法。

Physics understands matter through its atomic or molecular structure. This description has been overwhelmingly successful to predict physical properties and discover new materials. It is particularly well-suited to systems with a finite, though not too large, number of atoms and molecules. Phenomena occurring only when a large number of atoms or molecules interact call for other techniques. Phase transition is one such phenomenon: it is known that no phase transitions may occur unless the number of interacting objects tends to infinity. The proposed project aims at understanding the passage from finite systems (say, those with a finite number of atoms) to their continuum limit (the theories that assume from the start an infinite number of interacting objects). In particular it wants to identify the properties of finite systems that reveal those of the infinite ones obtained by increasing the number of objects. The main mathematical tool of the project will be algebra. Several chapters of mathematics have played a role in the description of phase transitions. Analysis and algebra are probably the central ones. Algebra provides the tools to identify the symmetries of physical systems. Symmetries are operations that transforms a system into another one without changing its overall physical properties. These symmetries offer fundamental ways to approach physical systems and define them mathematically. The proposed project puts an emphasis on studying the algebraic structures arising in both the finite lattice models and their infinite continuum limits. The research will focus on two-dimensional lattice models of microscopic interactions. These models are known as percolation, the Ising model, the XXZ spin chain, dense and dilute loop models, etc. They offer a natural laboratory to probe physical properties and prove them rigorously. They have a finite number of "particles", they can be probed on the computer and they are believed to go to (logarithmic) conformal field theories (a distinguished set of well-studied continuum models). Most importantly they rest upon an algebraic description that lends itself naturally to the study of the limit to large number of particles. Previous works in these directions, others' and mine, have contributed to both physics and mathematics. For example it is useful in physics to recognize emerging properties of a finite system, even though there are only partially realized. In mathematics, the study of algebraic structures of physical systems has suggested many new avenues of development or new ways of looking at existing results.