Algebraic Combinatorics

组合学处理离散对象，如有限集、图和置换。许多连续现象允许离散表示，使它们适合组合研究方法。本项目研究代数和几何中的组合结构，重点研究聚类代数理论和应用的进一步发展。聚类代数及其相关的组合和代数结构是在世纪之交被发现的。它们在其他数学和物理领域应用的重要性日益明显。在代数组合学中，常常会有相似的组合结构构成看似无关的数学现象。因此，隐藏的联系被揭示出来，允许见解和技术从一个学科转移到另一个学科。该项目旨在扩展和深化这些联系。该项目也将涉及研究生。这项研究受到几个经典数学领域的启发。主要的工具来自组合学，包括组合拓扑学、颤振突变机制和双色平面图的组合学。聚类代数及其基础的颤振突变组合学在许多数学学科中都有应用，包括表示理论、teichm<s:1> ller理论、数学物理、枚举和几何组合学。本项目提出了聚类理论在平面代数曲线以及奇点理论和低维拓扑等相关问题研究中的新应用。其他研究方向包括代数组合学在离散几何和计算几何中的应用、平面投影构型的研究和离散可积系统。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Combinatorics deals with discrete objects such as finite sets, graphs, and permutations. Many continuous phenomena allow for a discrete representation, lending themselves amenable to combinatorial methods of study. This project investigates combinatorial structures arising in algebra and geometry, focusing on further development of the theory and applications of cluster algebras. Cluster algebras and the associated combinatorial and algebraic constructions were discovered at the turn of the century. Their importance for applications in other areas of mathematics and physics is becoming increasingly apparent. In algebraic combinatorics, it is often the case that similar combinatorial structures underlie seemingly unrelated mathematical phenomena. As a result, hidden connections are revealed, allowing the transfer of insights and techniques from one discipline to another. This project aims to extend and deepen these connections. The project will also involve graduate students.This research is motivated by several classical areas of mathematics. The main tools come from combinatorics, including combinatorial topology, the machinery of quiver mutations, and combinatorics of bicolored planar graphs. Cluster algebras, and the combinatorics of quiver mutations underlying them, have found applications in many mathematical disciplines including representation theory, Teichmüller theory, mathematical physics, and enumerative and geometric combinatorics. The project suggests new applications of cluster theory to the study of plane algebraic curves and related problems of singularity theory and low-dimensional topology. Other research directions involve applications of algebraic combinatorics to discrete and computational geometry, the study of planar projective configurations, and discrete integrable systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.