Polyhedral Combinatorics in Representation Theory and Algebraic Geometry

该项目的重点是最近发现的集群代数的研究人员在合作S. Fomin。簇代数是一类特殊的整环，它为研究半单群中的全正性和正则基及其表示提供了一个简单的框架，本文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代数结构理论、热力学Bethe代数、quiver表象和可积系统等不同学科的意想不到的联系。该研究的主要工具之一是多面体组合学，更具体地说，是一种基于热带微积分的分段线性和无减法双有理变换之间的相互作用。该项目的主要动机来自数学的两个经典领域：表示论和整体正性理论。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（数组）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。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。

The project focuses on cluster algebras recently discovered by theinvestigator in collaboration with S.Fomin. Cluster algebras areintegral domains of a special kind designed to provide analgebraic framework for the study of total positivity andcanonical bases in semisimple groups and their representations.The investigator studies structural properties of cluster algebrasand their quantum deformations. This study uncovers unexpectedconnections with such diverse subjects as the structural theory ofKac-Moody algebras, thermodynamic Bethe ansatz, quiverrepresentations, and integrable systems. One of the maininstruments of the study is polyhedral combinatorics, morespecifically, an interplay between piecewise-linear andsubtraction-free birational transformations based on the tropicalcalculus.The main motivation for this project comes from two classical areasof mathematics: representation theory and the theory of totalpositivity. Representation theory is a mathematical approach tostudying symmetry; more specifically, it encodes the symmetryproperties of various physical and biological systems that occur innature. Total positivity is a remarkable property of matrices (arraysof numbers) that generalizes the familiar notion of positive numbers.Both theoriesfind numerous applications in physics, chemistry and othersciences, as well as in other mathematical disciplines. In fact,representation theory serves as the mathematical foundation ofquantum mechanics, while total positivity is a major tool forexplaining oscillations in mechanical systems. During the lastdecade, deep connections were found between the two fields, andthe scope of their applications was greatly extended. Thisproject explores the modern framework of representationtheory and total positivity, with the goal of making its formalismmuch more explicit and understandable.