Polyhedral combinatorics in representation theory and algebraic geometry

摘要主要研究者：Zelevinsky， Andrei提案号：DMS - 0801187机构：东北大学题目：表示理论和代数几何中的多面体组合研究重点是PI与S. Fomin合作发现的一类交换环——聚类代数。这一理论的出现是为了尝试为两个经典领域的研究创造一个代数框架：总正性理论和半单李群的表示理论。自成立以来，簇代数理论发现了许多令人兴奋的联系和应用：颤振表示、预投影代数、Calabi-Yau代数和范畴、Seiberg对偶、离散可积系统、泊松几何等。PI探讨了簇代数的结构性质，以及它们之间的联系和应用。他还在理论物理学中的超势理论的推动下，发展了具有势及其表征的颤振理论。研究的主要工具之一是多面体组合。这个项目植根于数学的两个经典领域：表示理论和总正性理论。表征理论是研究对称性的一种数学方法；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。总正性是矩阵（数字的正方形数组）的一个显著性质，它概括了熟悉的正数概念。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广泛的应用。事实上，表征理论是量子力学的数学基础，而总正性是解释力学系统振荡的主要工具。在过去的十年中，这两个领域之间发现了深刻的联系，它们的应用范围得到了极大的扩展。本项目探讨了表征理论和整体积极性的现代框架，目的是使其形式主义更加明确和可理解。

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.