Topological, Enumerative, and Algebraic Combinatorics

PI John Shareshian 研究在其他数学领域中出现或对其产生影响的组合数学问题。 组合学是对离散的、通常是有限的数学结构的研究。 这种结构经常出现在数学、自然科学和社会科学中，并产生各种应用。 例如，组合主义者将各种类型的网络建模为图，这些图只是点的集合，其中一些点对被认为是相关的。 尽管这些模型很简单，数学家们还是推导出了许多关于它们的深刻且适用的定理。 也许令人惊讶的是，组合数学与研究非离散对象的其他数学领域（包括拓扑和几何）之间存在密切联系。 PI Shareshian 的工作涉及对这种联系的密切研究，旨在解决有关离散和非离散结构的问题。PI Shareshian 研究组合数学与其他数学领域（包括代数、拓扑和几何）之间的联系。 在与 Michelle Wachs 的合作中，他的目标是证明单位区间图的 Stanley 色对称函数的改进版本的分级 Frobenius 特性实际上是相关正则半单 Hessenberg 簇的上同调，并利用这样的结果来攻击 Stanley 和 Stembridge 关于此类对称函数的长期猜想。 Shareshian 研究结构代数对象和与此类对象自然关联的格子组合结构之间的联系。 例如，他的目标是提供一个关于李代数的结果，该结果与他之前的结果大致相似，即当且仅当其子群格的阶复数是可壳的时，有限群才是可解的。

PI John Shareshian studies problems in combinatorics that arise in or have consequences for other fields of mathematics. Combinatorics is the study of discrete, typically finite, mathematical structures. Such structures arise often in mathematics and the natural and social sciences, leading to various applications. For example, networks of various types are modeled by combinatorialists as graphs, which are simply collections of points, some pairs of which are considered to be related. Despite the simplicity of such models, mathematicians have derived many deep and applicable theorems about them. Perhaps surprisingly, there are close connections between combinatorics and other fields of mathematics in which non-discrete objects are studied, including topology and geometry. The work of PI Shareshian involves the close study of such connections, with the aim of solving problems about both discrete and non-discrete structures.PI Shareshian studies connections between combinatorics and other fields of mathematics, including algebra, topology, and geometry. In joint work with Michelle Wachs, he aims to prove that the graded Frobenius characteristic of a refined version of Stanley's chromatic symmetric function for a unit interval graph is in fact the cohomology of an associated regular semisimple Hessenberg variety, and to use such a result to attack a longstanding conjecture of Stanley and Stembridge about such symmetric functions. Shareshian studies connections between the structure algebraic objects and the combinatorial structure of lattices naturally associated to such objects. For example, he aims to provide a result on Lie algebras roughly analogous to his earlier result that a finite group is solvable if and only if the order complex of its subgroup lattice is shellable.