Algebraic Combinatorics

这个项目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、代数组合学和箭图突变机制。组合学处理离散对象，如有限集、图、排列、偏序等。许多连续现象允许离散表示，使它们本身服从于组合研究方法。通常的情况是，相同或相似的组合结构构成了看似不相关的数学实体的基础，揭示了这些实体之间的隐藏联系，并允许将洞察力和技术从一个学科转移到另一个学科。一个很好的例子是簇代数理论，这是这个项目的主要焦点。这个奖项支持了PI在代数和几何中对组合结构的研究，重点是簇代数的理论和应用。簇代数是由PI和A.Zlevinsky合作发现的，它已经在几个数学学科中得到了应用，包括表示论、Teichmueller理论、数学物理、计数和几何组合学。作者打算进一步探索簇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受限算术运算集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.