Polyhedral Combinatorics in Representation Theory and Algebraic Geometry

该项目的重点是集群代数发现的研究者在合作S. Fomin。簇代数是一类特殊的交换代数，它为研究半单群的全正性和标准基及其表示提供了一个代数框架。研究了团簇代数的结构性质及其量子形变.这项研究揭示了意想不到的连接suchdifferent科目的结构理论的卡茨-穆迪代数，表示理论的有限维代数，几何ofmoduli空间，和理论的超势。研究的主要工具之一是多面体组合学。 这个项目的动机是两个经典的数学领域：表示理论和理论的总积极性。表示理论是一种数学方法来研究对称性;更具体地说，它编码的对称性在自然界中发生的各种物理和生物系统。全正性是矩阵（数字方阵）的一个显着属性，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许多应用。事实上，表象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。近十年来，这两个领域之间的联系越来越紧密，应用范围也越来越广。这个项目探讨了表征理论和总体积极性的现代框架，目的是使其形式主义更加明确和易于理解。

The project focuses on cluster algebras discovered by theinvestigator in collaboration with S.Fomin. Cluster algebras arecommutative of a special kind designed to provide an algebraicframework for the study of total positivity and canonical bases insemisimple groups and their representations. The investigator studiesstructural properties of cluster algebras and their quantumdeformations. This study uncovers unexpected connections with suchdiverse subjects as the structural theory of Kac-Moody algebras,representation theory of finite dimensional algebras, geometry ofmoduli spaces, and the theory of superpotentials. One of the maininstruments of the study is polyhedral combinatorics. This project is motivated by 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 variousphysical and biological systems that occur in nature. Total positivityis a remarkable property of matrices (square arrays of numbers) thatgeneralizes the familiar notion of positive numbers. Both theoriesfind numerous applications in physics, chemistry and other sciences,as well as in other mathematical disciplines. In fact, representationtheory serves as the mathematical foundation of quantum mechanics,while total positivity is a major tool for explaining oscillations inmechanical systems. During the last decade, deep connections werefound between the two fields, and the scope of their applications wasgreatly extended. This project explores the modern framework ofrepresentation theory and total positivity, with the goal of makingits formalism more explicit and understandable.