Enumerative, Algebraic and Topological Combinatorics

约翰Shareshian工程的问题，代数，拓扑和计数组合数学有密切的联系，与其他领域的数学。 单独，在联合工作与米歇尔瓦克斯和additialjoint工作与菲利普汉隆和帕特里夏赫什，他研究组actionson同源的秩序复杂的各种偏序集，与应用置换枚举和代表性理论的对称群和有限群的李型。 在与DavidWright的合作中，他研究了某些向量空间，其基础由有限树索引，这是在雅可比猜想的研究中出现的。 粗略地说，组合学是研究离散的，通常是有限的，连续的对象。 组合问题自然地出现在各种科学学科中，包括生物学、计算机科学和电子工程，以及纯数学的大多数领域。 大部分的工作特别支持这项补助金涉及对称性的研究偏序集。 偏序集是一个集合，其中一些（但不一定是所有）元素对a，B以模仿真实的数集合上的关系ab的方式相关。 (For例如，如果a、B相关，而B、c相关，则a、c必须相关。） 偏序集合的不对称性是集合的重排，它将相关的对发送到相关的对。 (For例如，如果我们取所有具有通常关系的真实的数的集合，则通过将每个数x发送到x+1所得到的重排是对称的，而通过将x发送到-x所得到的重排不是。 偏序集和它们的对称性是数学家在纯数学的几个领域中所感兴趣的，包括拓扑学和代数学。 它们被用来解决理论计算机科学中的难题。

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.