Algebraic Combinatorics

该项目主要研究代数和几何中出现的组合结构，重点是簇代数的理论和应用。簇代数是与A.Zlevinsky合作发现的一类交换环，它在表示论、TeichMueller理论、离散动力系统、全正性、Lie理论、热带几何以及计数和几何组合学等数学学科中都有应用。研究人员发展了簇代数的一般结构理论和相关的组合结构，并将其应用于研究在各种应用中出现的具体的簇代数类。主要工具来自组合学，包括组合拓扑学、代数和几何组合学以及根系理论。组合学处理离散对象，如有限集、图、排列、偏序等。许多连续现象允许离散表示，使它们本身服从于组合研究方法。通常的情况是，相同或相似的组合结构构成了紧密不相关的数学实体的基础，揭示了它们之间的隐藏联系，并允许将洞察力和技术从一个学科传递到另一个学科。簇代数理论就是一个很好的例子，它是这个项目的主要焦点。

The project focuses on the study of combinatorial structures arisingin algebra and geometry, with an emphasis on the theory and applicationsof cluster algebras. Cluster algebras, discovered by the investigator incollaboration with A.Zelevinsky, are a class of commutative rings whichhave found applications in several mathematical disciplines, includingrepresentation theory, Teichmueller theory, discrete dynamical systems,total positivity, Lie theory, tropical geometry, and enumerative andgeometric combinatorics. The investigator develops general structuraltheory of cluster algebras and related combinatorial constructions, andapplies it to the study of concrete classes of cluster algebras arisingin various applications.The original motivation for this project comes from several classicalareas of mathematics listed above. The main tools come from combinatorics,including combinatorial topology, algebraic and geometric combinatorics, and the theory of root systems. Combinatorics deals with discrete objects such as finite sets, graphs, permutations, partial orders, etc. Many continuous phenomena allow for a discrete representation, lending themselves amenable to combinatorial methods of study. It is often the case that identical or similar combinatorial structures underlie seamingly unrelated mathematical entities, revealing hidden connections between them and allowing to transport insights and techniques from one discipline to another. One case in point is the theory of cluster algebras, which are the main focus of this project.