Enumerative, Algebraic and Topological Combinatorics

John Shareshian研究与其他数学领域有密切联系的代数、拓扑和枚举组合问题。在与Michelle Wachs以及Philip Hanlon和Patricia Hersh的共同研究中，他研究了各种偏序集合的有序复合体的群作用同调，并将其应用于对称群和李型有限群的置换枚举和表示理论。在与DavidWright的合作中，他研究了在雅可比猜想中出现的以有限树为索引的向量空间。这项奖助金支持组合学领域的工作。粗略地说，组合学是研究离散的，通常是有限的数学对象。组合问题自然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，以及大多数纯数学领域。该基金专门支持的大部分工作涉及对部分有序集的对称性的研究。部分有序集合是这样一种集合，其中一些（但不一定是全部）元素对A，b以类似实数集合上的关系ab的方式相关联。（例如，如果a，b是相关的，b，c是相关的，那么a，c一定是相关的。）偏序集合的不对称性是指该集合将相关对发送到相关对的重排。（例如，如果我们取具有通常关系的所有实数的集合，那么通过将每个数字x发送给x+1获得的重排是对称的，而将x发送给-x获得的重排则不是对称的。）部分有序集及其对称性对研究纯数学的几个领域的数学家很感兴趣，包括拓扑和代数。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.