Algebraic Enumerative and Topological Combinatorics

John Shareshian计划从事四个项目。他将继续与Michelle Wachs合作，在各种组合、几何和表示理论设置中使用准对称函数。该项目目前的主要目标是证明正则半简单Hessenberg变体上同调上对称群的表示的Frobenius特征是与该变体自然相关的图的色对称函数的改进。Shareshian将继续与Russ Woodroofe合作，研究两个问题。第一个问题是证明有限群的余集偏集的复阶永不可缩并。Shareshian和Woodroofe将这一问题简化为一个关于交替群的极大子群的指标的素因子问题。第二个问题是发展理查德·斯坦利的超可解晶格理论的非分级模拟。主要目标是找到一种理论，既包含已经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。最后，Shareshian将尝试解决Michael Aschbacher和Stephen Smith在关于有限群的p-子群的偏序集的Quillen猜想的工作中提出的拓扑组合中的一个问题。Shareshian研究了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，通常试图列举所有满足给定条件的结构。组合问题自然出现在智力探究的各个领域，包括数学、计算机科学、物理学和生物学。群论通过一个称为群的代数系统对物体的对称性进行编码来研究对称性。由于人类自然地被高度对称的物体所吸引，并利用对称来理解许多科学和美学现象，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。这些形状可能存在于高维空间中，因此无法通过可视化来理解。数学家经常试图通过将代数系统与这些形状联系起来，以提供有用的信息来理解它们。在某些情况下，通过将与手头几何问题相关的代数信息编码的组合对象与这些代数系统相关联，可以更好地理解这些代数系统。Shareshian研究以这种方式产生的组合对象。

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.