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.
