Algebraic Combinatorics

该项目主要研究代数和几何中的组合结构，重点是簇代数的理论和应用。由研究者与A.Zelevinsky合作发现的簇代数，已经在几个数学学科中得到了应用，包括表示理论、Teichmueller理论、离散动力系统、离散势理论、李理论、热带几何、枚举和几何组合学。研究者发展了聚类代数和相关组合结构的一般结构理论，并将其应用于各种应用中出现的具体类聚类代数的研究。这个项目的动机是由上面列出的几个经典的数学领域。主要的工具来自组合学，包括组合拓扑学、代数和几何组合学，以及颤振突变机制。组合学处理离散对象，如有限集、图、置换、偏序等。许多连续现象允许离散表示，使它们适合组合研究方法。通常情况下，相同或相似的组合结构隐藏在不相关的数学实体之下，揭示了它们之间隐藏的联系，并允许将见解和技术从一个学科转移到另一个学科。一个恰当的例子是聚类代数理论，这是本项目的主要焦点。

The project focuses on the study of combinatorial structures arising in algebra and geometry, with an emphasis on the theory and applications of cluster algebras. Cluster algebras, discovered by the investigator in collaboration with A.Zelevinsky, have found applications in several mathematical disciplines, including representation theory, Teichmueller theory, discrete dynamical systems, discrete potential theory, Lie theory, tropical geometry, and enumerative and geometric combinatorics. The investigator develops general structural theory of cluster algebras and related combinatorial constructions, and applies it to the study of concrete classes of cluster algebras arising in various applications.This project is motivated by several classical areas of mathematics listed above. The main tools come from combinatorics, including combinatorial topology, algebraic and geometric combinatorics, and the machinery of quiver mutations. 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 transfer insights and techniques from one discipline to another. A case in point is the theory of cluster algebras, which are the main focus of this project.