Algebraic Combinatorics

组合学处理离散对象，如有限集，图和排列。许多连续的现象允许离散的表示，贷款自己服从组合的研究方法。本计画研究代数与几何中出现的组合结构，著重于群代数理论与应用的进一步发展。在世纪之交，人们发现了簇代数及其相关的组合和代数结构。它们在数学和物理的其他领域的应用的重要性正变得越来越明显。在代数组合学中，通常情况下，相似的组合结构隐藏在看似无关的数学现象之下。因此，隐藏的联系被揭示出来，允许将见解和技术从一个学科转移到另一个学科。该项目旨在扩大和深化这些联系。研究生也将参与该项目，这项研究的动机是数学的几个经典领域。主要的工具来自组合学，包括组合拓扑学，机器的突变，和组合学的双色平面图。簇代数和它们的基础上的突变的组合学在许多数学学科中有应用，包括表示论，Teichmüller理论，数学物理，枚举和几何组合学。该项目提出了新的应用集群理论的研究平面代数曲线和相关问题的奇异性理论和低维拓扑。其他研究方向包括代数组合学在离散和计算几何中的应用、平面投影配置的研究和离散可积系统。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.