Cluster algebras: theory and applications

簇代数：理论与应用

• 批准号: 1103813
• 负责人: Gordana Todorov
• 金额: $ 30万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1103813&HistoricalAwards=false
• 关键词: Cluster; algebras; theory; applications

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.

建议的研究集中在簇代数上，这是PI与S.Fomin合作发现的一类交换环。这一理论的提出是为了建立一个代数框架来研究两个经典领域：全正性理论和半单李群的表示理论。自从它诞生以来，簇代数理论发现了许多令人兴奋的联系和应用：箭图表示，非交换几何，Seiberg对偶，离散可积系统，TeichMuller理论等。PI探索了簇代数的结构性质，以及它们的联系和应用。在理论物理中的超势理论的启发下，他还发展了有势箭图及其表示的理论。这个项目植根于数学的两个经典领域：表象理论和全面性理论。表示论为研究对称性提供了数学工具，而全正性是矩阵(数的平方数组)的一个显著性质，它推广了人们熟悉的正数概念。这两个理论在物理、化学和其他科学中都有大量的应用，也与其他数学学科有许多联系。事实上，表象理论是量子力学的数学基础，而全正性是解释机械系统振荡的主要工具。位于本项目核心的簇代数为这些学科的研究提供了一个新的代数框架，使它们的形式更加明确和易于理解。