Topological, Enumerative, and Algebraic Combinatorics

Pi John Shareshian研究组合学中的问题，这些问题出现在其他数学领域，或对其他数学领域产生影响。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，导致了各种应用。例如，组合学家将各种类型的网络建模为图，这些图仅仅是点的集合，其中一些点对被认为是相关的。尽管这些模型很简单，但数学家们已经得到了许多关于它们的深刻而适用的定理。也许令人惊讶的是，组合学和其他研究非离散对象的数学领域之间有着密切的联系，包括拓扑学和几何学。Pi Shareshian的工作涉及对这种联系的密切研究，目的是解决离散和非离散结构的问题。PI Shareshian研究组合学和其他数学领域之间的联系，包括代数、拓扑和几何。在与Michelle Waches的合作中，他的目的是证明单位区间图的Stanley色对称函数的精化形式的分次Frobenius特征实际上是相关的正则半单Hessenberg簇的上同调，并利用这样的结果来攻击Stanley和Stembridge关于这种对称函数的一个长期猜想。Shareshian研究了结构代数对象和与这些对象自然相关的格子的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，该结果与他早期的结果大致相似，即一个有限群是可解的当且仅当它的子群格序复形是可壳的。

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.