Polyhedral Combinatorics in Representation Theory and Algebraic Geometry

该项目的重点是研究人员与S.Fomin合作发现的簇代数。簇代数是一种特殊的交换性，旨在为研究半单群及其表示中的全正性和标准基提供一个代数框架。研究了簇代数的结构性质及其量子形变。这项研究揭示了与Kac-Moody代数的结构理论、有限维代数的表示理论、模空间的几何和超势理论等不同学科的意想不到的联系。研究的主要工具之一是多面体组合数学。这个项目是由两个经典的数学领域推动的：表象理论和全正理论。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。这两个理论在物理、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，这两个领域之间建立了深刻的联系，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.