Algebraic Enumerative and Topological Combinatorics

John Shareshian计划参与四个项目。他将继续与米歇尔·瓦克斯合作，涉及在各种组合、几何和表示理论环境中使用准对称函数。这个项目目前的主要目标是证明正则半单Hessenberg簇的上同调上的对称群的表示的Frobenius特征是与该簇自然相关的图的色对称函数的精化。沙雷希安将继续与拉斯·伍德罗夫合作，其中研究了两个问题。第一个问题是证明有限群的陪集偏序集的序复形永远不是可压缩的。Shareshian和Woodroofe将这个问题归结为交错群的极大子群的指数的素因子问题。第二个问题是发展理查德·斯坦利的超可解格理论的非分级类比。其主要目的是找到一种既包含已被充分研究的左模格理论，又能更好地理解有限可解群的子群格的组合性质的理论。最后，Shareshian将尝试解决Michael Aschbacher和Stephen Smith在他们关于有限群的p-子群的偏序集的Quillen猜想工作中提出的拓扑组合学中的一个问题。Shareshian研究三个数学领域之间的联系，即组合学、群论和几何。组合学家研究离散的，通常是有限的结构，经常试图列举满足给定条件的所有这样的结构。组合问题自然而然地出现在智力研究的各个领域，包括数学、计算机科学、物理和生物。群论者通过用称为群的代数系统来编码对象的对称性来研究对称性。由于人类天生就被高度对称的物体所吸引，并利用对称性来理解许多科学和美学现象，因此群论无处不在。例如，它出现在物理和化学的许多问题中。几何学家研究形状。这些形状可能生活在高维空间中，因此无法通过可视化来理解。数学家经常试图通过将提供有用信息的代数系统与它们联系起来来理解这些形状。在某些情况下，通过将编码与手边几何问题相关的代数信息的组合对象与这些代数系统相关联来最好地理解这些代数系统。Shareshian研究了以这种方式出现的组合对象。

John Shareshian plans to work on four projects. He will continue joint work with Michelle Wachs involving the use of quasisymmetric functions in various combinatorial, geometric and representation theoretic settings. The current main goal of this project is to show that the Frobenius characteristic of a representation of the symmetric group on the cohomology of a regular semisimple Hessenberg variety is a refinement of the chromatic symmetric function of a graph naturally associated to the variety. Shareshian will continue joint work with Russ Woodroofe, in which two problems are studied. The first problem is to show that the order complex of the coset poset of a finite group is never contractible. This problem has been reduced by Shareshian and Woodroofe to a problem about prime divisors of indices of maximal subgroups of alternating groups. The second problem is to develop a nongraded analog of Richard Stanley's theory of supersolvable lattices. The main goal is to find a theory that both encompasses the already well-studied theory of left modular lattices and provides a better understanding of combinatorial properties of subgroup lattices of finite solvable groups. Finally, Shareshian will attempt to settle a problem in topological combinatorics raised by Michael Aschbacher and Stephen Smith in their work on the Quillen Conjecture about the partially ordered set of p-subgroups of a finite group.Shareshian studies connections between three areas of mathematics, namely, combinatorics, group theory and geometry. Combinatorialists study discrete, usually finite, structures, often attempting to enumerate all such structures satisfying a given condition. Combinatorial problems arise naturally in various areas of intellectual inquiry, including mathematics, computer science, physics and biology. Group theorists study symmetry by encoding the symmetries of an object by an algebraic system, called a group. Since humans are naturally drawn to highly symmetric objects and use symmetry to understand many scientific and aesthetic phenomena, group theory is ubiquitous. It appears, for example, in many problems from physics and chemistry. Geometers study shapes. These shapes might live in spaces of high dimension and therefore be inaccessible to understanding through visualization. Mathematicians often attempt to understand such shapes by associating to them algebraic systems in a way that provides useful information. In some cases, these algebraic systems are best understood by associating to them combinatorial objects that encode the algebraic information relevant to the geometric problem at hand. Shareshian examines combinatorial objects arising in this manner.