Enumerative, Algebraic and Topological Combinatorics

Washington University, Saint Louis MO

Investigators

Abstract

John Shareshian works on problems in algebraic, topological and enumerative combinatorics that have close connections with other areas of mathematics. Alone, in joint work with Michelle Wachs and in additional joint work with Philip Hanlon and Patricia Hersh, he studies group actions on homology of order complexes of various partially ordered sets, with applications to permutation enumeration and representation theory of symmetric groups and finite groups of Lie type. In joint work with David Wright, he studies certain vector spaces with bases indexed by finite trees, which arise in the study of the Jacobian Conjecture. This grant supports work in the area of combinatorics. Roughly, combinatorics is the study of discrete, usually finite, mathematical objects. Combinatorial problems arise naturally in various scientific disciplines, including biology, computer science and electrical engineering, and in most areas of pure mathematics. Most of the work specifically supported by this grant involves the study of symmetries of partially ordered sets. A partially ordered set is a set in which some (but not necessarily all) pairs a,b of elements are related in a manner that mimics the relation a<b on the set of real numbers. (For example, if a,b are related and b,c are related then a,c must be related.) A symmetry of a partially ordered set is a rearrangement of the set which sends related pairs to related pairs. (For example, if we take the set of all real numbers with the usual < relation, then the rearrangement obtained by sending each number x to x+1 is a symmetry, while the rearrangement obtained by sending x to -x is not.) Partially ordered sets and their symmetries are of interest to mathematicians working in several areas of pure mathematics, including topology and algebra. They have been used to attack difficult problems in theoretical computer science.