Enumerative, Algebraic and Topological Combinatorics

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John Shareshian works on problems in algebraic, topological andenumerative combinatorics that have close connections with other areas ofmathematics. Alone, in joint work with Michelle Wachs and in additionaljoint work with Philip Hanlon and Patricia Hersh, he studies group actionson homology of order complexes of various partially ordered sets, withapplications to permutation enumeration and representation theory ofsymmetric groups and finite groups of Lie type. In joint work with DavidWright, he studies certain vector spaces with bases indexed by finitetrees, 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, mathematicalobjects. Combinatorial problems arise naturally in various scientificdisciplines, including biology, computer science and electricalengineering, and in most areas of pure mathematics. Most of the workspecifically supported by this grant involves the study of symmetries ofpartially 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 mannerthat mimics the relation ab on the set of real numbers. (For example,if a,b are related and b,c are related then a,c must be related.) Asymmetry of a partially ordered set is a rearrangement of the set whichsends related pairs to related pairs. (For example, if we take the set ofall real numbers with the usual relation, then the rearrangementobtained by sending each number x to x+1 is a symmetry, while therearrangement obtained by sending x to -x is not.) Partiallyordered sets and their symmetries are of interest to mathematiciansworking in several areas of pure mathematics, including topology andalgebra. They have been used to attack difficult problems in theoreticalcomputer science.