Algebraic Combinatorics

这个项目研究了代数和几何中的组合结构，包括簇代数理论和应用的进一步发展。组合学处理离散的对象，如有限集合、图和排列；许多连续现象允许离散的表示，使它们本身也服从于组合研究方法。簇代数和作为其基础的组合学是最近才被发现的，但它们在数学和物理的其他领域中的应用的重要性正变得越来越明显。通常的情况是，类似的组合结构构成了看似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转移到另一个学科。这个项目旨在扩展和深化这些联系。这项研究是由几个经典的数学领域推动的。主要工具来自组合学，包括组合拓扑学、箭图突变机制、对称函数和Young表以及组合环理论。簇代数，以及它们背后的箭图突变的组合学，已经在几个数学学科中找到了应用，包括表示论，Teichmüler理论，数学物理，计数和几何组合学。本文将研究簇论在平面代数曲线孤立奇点研究中的新应用。另一个研究方向是非对易Schur函数理论和组合环论的相关问题。该项目还将研究代数组合学在平面射影几何和代数复杂性理论中的应用。

This project investigates combinatorial structures arising in algebra and geometry, including further development of the theory and applications of cluster algebras. 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 as well. Cluster algebras and the combinatorics underlying them were discovered only relatively recently, but their importance for application in other areas of mathematics and physics is becoming increasingly apparent. It is often the case that similar combinatorial structures underlie seemingly unrelated mathematical entities, revealing hidden connections between them and allowing the transfer of insights and techniques from one discipline to another. This project aims to extend and deepen these connections.This research is motivated by several classical areas of mathematics. The main tools come from combinatorics, including combinatorial topology, the machinery of quiver mutations, symmetric functions and Young tableaux, and combinatorial ring theory. Cluster algebras, and the combinatorics of quiver mutations underlying them, have found applications in several mathematical disciplines including representation theory, Teichmüller theory, mathematical physics, and enumerative and geometric combinatorics. A new application of cluster theory to the study of isolated singularities of plane algebraic curves will be studied. Another research direction concerns the theory of noncommutative Schur functions and related questions of combinatorial ring theory. The project will also investigate applications of algebraic combinatorics to planar projective geometry and algebraic complexity theory.