Polyhedral combinatorics in representation theory and algebraic geometry

主要研究者：Zelevinsky，Andrei 提案编号：DMS -0801187机构：东北大学题目：表示论和代数几何中的多面体组合学拟议的研究重点是簇代数，这是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，簇代数的理论发现了一些令人兴奋的连接和应用程序：群表示，预投射代数，Calabi-Yau代数和范畴，Seiberg对偶，离散可积系统，泊松几何等PI探索簇代数的结构特性，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。研究的主要工具之一是多面体组合数学。这个项目植根于数学的两个经典领域：表示论和全正性理论。表示论是一种研究对称性的数学方法;更具体地说，它编码了自然界中各种物理和生物系统的对称性。全正性是矩阵（数字的方阵）的一个显着属性，它推广了熟悉的正数概念。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许多应用。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。在过去的十年中，这两个领域之间的联系被发现，其应用范围大大扩展。这个项目探讨表征理论和总积极性的现代框架，其目标是使其形式主义更加明确和易于理解。

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.