Topological, Enumerative, and Algebraic Combinatorics

PI John Shareshian研究组合学中的问题，这些问题在其他数学领域中出现或对其他数学领域产生影响。组合学是研究离散的，通常是有限的数学结构。这种结构经常出现在数学、自然科学和社会科学中，导致各种各样的应用。例如，组合学家将各种类型的网络建模为图，图是点的简单集合，其中一些点对被认为是相关的。尽管这些模型很简单，但数学家们已经推导出了许多深刻而适用的定理。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的工作涉及对这种连接的密切研究，目的是解决有关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 Wachs的合作中，他旨在证明Stanley的单位间隔图的色对称函数的改进版本的梯度Frobenius特征实际上是相关的正则半简单Hessenberg变体的上同调，并利用这一结果来攻击Stanley和Stembridge关于这种对称函数的长期猜想。Shareshian研究结构代数对象和与这些对象自然相关的格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，大致类似于他之前的结果，即当且仅当子群格的阶复是可壳的，有限群是可解的。

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.