Algebraic Combinatorics

这个项目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、代数组合学和颤振突变机制。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看似无关的数学实体之下，揭示了这些实体之间的隐藏联系，并允许从一个学科到另一个学科的见解和技术转移。一个恰当的例子是聚类代数理论，这是本项目的主要焦点。该奖项支持PI对代数和几何中出现的组合结构的研究，重点是聚类代数的理论和应用。聚类代数是PI与A. Zelevinsky合作发现的，已经在几个数学学科中得到了应用，包括表示理论、Teichmueller理论、数学物理、枚举和几何组合学。研究者打算进一步探索聚类代数与经典不变理论和射影几何之间的联系。另一个研究方向涉及使用一组有限算术运算的计算模型的代数复杂性。

This project is motivated by several classical areas of mathematics. The main tools come from combinatorics, including combinatorial topology, algebraic 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 seemingly unrelated mathematical entities, revealing hidden connections between these and allowing the transfer of 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.This award supports the PI's research on combinatorial structures arising in algebra and geometry, with an emphasis on the theory and applications of cluster algebras. Cluster algebras, discovered by the PI in collaboration with A. Zelevinsky, have found applications in several mathematical disciplines including representation theory, Teichmueller theory, mathematical physics, and enumerative and geometric combinatorics. The investigator intends to further explore the connections between cluster algebras, on one hand, and classical invariant theory and projective geometry, on the other. Another research direction concerns algebraic complexity for computational models employing a restricted set of arithmetic operations.