Polyhedral combinatorics in representation theory and algebraic geometry

表示论和代数几何中的多面体组合

• 批准号: 0801187
• 负责人: Andrei Zelevinsky
• 金额: $ 20.1万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801187&HistoricalAwards=false
• 关键词: Polyhedral; combinatorics; representation; theory; algebraic

ABSTRACTPrincipal Investigator: Zelevinsky, Andrei Proposal Number: DMS - 0801187Institution: Northeastern UniversityTitle: Polyhedral combinatorics in representation theory and algebraic geometryThe 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, preprojective algebras, Calabi-Yau algebras and categories, Seiberg dualities, discrete integrable systems, Poisson geometry, 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. One of the main instruments of the study is polyhedral combinatorics.This project has roots in two classical areas of mathematics: representation theory and the theory of total positivity. Representation theory is a mathematical approach to studying symmetry; more specifically, it encodes the symmetry properties of various physical and biological systems that occur in nature. 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 in 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. During the last decade, deep connections were found between the two fields, and the scope of their applications was greatly extended. This project explores the modern framework of representation theory and total positivity, with the goal of making its formalism more explicit and understandable.

主要研究人员：Zlevinsky，Andrei提案编号：DMS-0801187机构：东北大学标题：表示理论和代数几何中的多面体组合学建议的研究重点是簇代数，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。自创立以来，簇代数理论发现了许多令人兴奋的联系和应用：箭图表示、预投射代数、Calabi-Yau代数和范畴、Seiberg对偶、离散可积系统、Poisson几何等。PI探索了簇代数的结构性质及其联系和应用。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。这项研究的主要工具之一是多面体组合学。这个项目植根于两个经典的数学领域：表示论和全正性理论。表象理论是一种研究对称性的数学方法；更具体地说，它编码了自然界中出现的各种物理和生物系统的对称性。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。这两种理论在物理、化学和其他科学以及其他数学学科中都有大量的应用。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。在过去的十年里，这两个领域之间发现了深刻的联系，它们的应用范围得到了极大的扩展。这个项目探索了表征理论和完全实证性的现代框架，目的是使其形式主义更明确和更容易理解。