Cluster algebras: theory and applications

拟议的研究集中在簇代数，一类交换环发现的PI与S。佛明这个理论的出现是为了试图为两个经典领域的研究建立一个代数框架：全正性理论和半单李群的表示理论。自成立以来，集群代数的理论发现了一些令人兴奋的连接和应用：群表示，非交换几何，Seiberg对偶，离散可积系统，Teichmuller理论等PI探索集群代数的结构特性，以及它们的连接和应用。他还发展了理论的颤抖与潜力和他们的代表性，动机除其他外，理论物理学的超势理论。这个项目源于数学的两个经典领域：表征理论和完全积极性理论。表示论为研究对称性提供了数学工具，而全正性是矩阵（数字的方阵）的一个显着属性，它概括了熟悉的正数概念。这两个理论在物理学、化学和其他科学中有许多应用，也与其他数学学科有许多联系。事实上，表示论是量子力学的数学基础，而全正性是解释力学系统振荡的主要工具。在这个项目的心脏躺在集群代数为这些学科的研究提供了一个新的代数框架，使他们的形式主义更加明确和易于理解。

The proposed research focuses on cluster algebras, a class of commutative rings discovered by the PI in collaboration with S. Fomin. This theory arose as an attempt to create an algebraic framework for the study of two classical fields: theory of total positivity, and representation theory of semisimple Lie groups. Since its inception, the theory of cluster algebras found a number of exciting connections and applications: quiver representations, non-commutative geometry, Seiberg dualities, discrete integrable systems, Teichmuller theory, etc. The PI explores the structural properties of cluster algebras, and their connections and applications. He also develops the theory of quivers with potentials and their representations, motivated among other things, by the theory of superpotentials in theoretical physics. This project has roots in two classical areas of mathematics: representation theory and the theory of total positivity. Representation theory provides mathematical tools for studying symmetry, while total positivity is a remarkable property of matrices (square arrays of numbers) that generalizes the familiar notion of positive numbers. Both theories find numerous applications in physics, chemistry and other sciences, as well as numerous connections with other mathematical disciplines. In fact, representation theory serves as the mathematical foundation of quantum mechanics, while total positivity is a major tool for explaining oscillations in mechanical systems. The cluster algebras lying in the heart of this project provide a new algebraic framework for the study of these disciplines, making their formalism more explicit and understandable.