Algebraic Combinatorics

本计画研究代数与几何中出现的组合结构，包括进一步发展丛集代数的理论与应用。 组合学处理离散对象，如有限集，图和排列;许多连续现象允许离散表示，也适合组合研究方法。 簇代数和它们背后的组合学只是最近才被发现，但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。通常情况下，相似的组合结构隐藏在看似无关的数学实体之下，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。本项目旨在扩展和深化这些联系。本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，机械的突变，对称函数和杨tableaux，和组合环理论。簇代数和它们的基础上的突变的组合学在几个数学学科中有应用，包括表示论，Teichmüller理论，数学物理，枚举和几何组合学。本文将研究丛理论在研究平面代数曲线孤立奇点中的一个新应用。另一个研究方向涉及非交换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.