Algebraic, topological and enumerative combinatorics

Shareshian 将致力于解决枚举、代数和拓扑组合学中的几个问题。 他将继续与 M. Wachs 合作开展一个涉及拟对称函数和排列统计的项目。 他将尝试扩大与 A. Hultman、S. Linusson 和 J. Sjostrand 的合作，涉及超平面安排和 Bruhat 订单。 他将继续尝试证明存在某个有限格L，使得不存在其子群格包含与L同构的区间的有限群G。特别地，他将检查出现在一个猜想中的一类格，该猜想既专门化了他自己的拓扑猜想，又专门化了M. Aschbacher的组合猜想。Shareshian在这个项目上的大部分工作在于三个数学领域的交叉，即组合学、拓扑学和有限群论。 重点是组合学，它是对具有一些附加结构的离散（通常是有限）集合的研究。通常，组合主义者希望知道存在多少给定类型的结构，或者有多少这样的结构拥有一些有趣的附加属性。 例如，考虑数字 1,...,n 的所有可能列表的集合 S(n)。 当n=3时，这样的列表有3x2x1=6个，即123,132,213,231,312,321。 一般来说，有 x(n-1)x(n-2)...x2x1 这样的列表，当 n 变得很大时，即使使用一台非常强大的计算机，也无法通过检查所有列表来回答有关这些列表的问题。 感兴趣的一类问题是，对于给定的 n 和 k，计算出 S(n) 中有多少成员恰好在列表中具有 k 个位置 i，其中位置 i 中的数字大于 i。 这个特定问题的答案很好理解，但 Shareshian 和 Michelle Wachs 正在研究各种概括。 Shareshian 与 Axel Hultman、Svante Linusson 和 Jonas Sjostrand 一起证明了 Alexander Postnikov 的猜想，该猜想将上述类型的问题与涉及将高维空间切成碎片的问题相关联。 他计划致力于推广这一结果。 最后，Shareshian 正在努力证明他自己和 Michael Aschbacher 提出的一个猜想，该猜想涉及一类代数对象（在纯数学和应用数学中普遍存在）（称为有限群）与一类或组合对象（称为有限格）。 在所有项目中，希望所考虑问题的解决方案以及用于获得这些问题的方法对于其他领域的组合学家和数学家来说都是有用和感兴趣的。

Shareshian will work on several problems in enumerative, algebraic and topological combinatorics. He will continue his joint work with M. Wachs on a project involving quasisymmetric functions and permutation statistics. He will try to extend joint work with A. Hultman, S. Linusson and J. Sjostrand involving hyperplane arrangements and Bruhat order. He will continue to try to prove that there is some finite lattice L such that there is no finite group G whose subgroup lattice contains an interval isomorphic with L. In particular, he will examine a class of lattices appearing in a conjecture that specializes both his own topological conjecture and a combinatorial conjecture of M. Aschbacher.Most of Shareshian's work on this project lies in the intersection of three areas of mathematics, namely, combinatorics, topology and finite group theory. The emphasis is on combinatorics, which is the study of discrete, usually finite, sets endowed with some additional structure. Typically, a combinatorialist wishes to know how many structures of a given type exist, or how many such structures possess some interesting additional property. For example, consider the set S(n) of all possible lists of the numbers 1,...,n. When n=3, there are 3x2x1=6 such lists, namely, 123,132,213,231,312,321. In general, there arenx(n-1)x(n-2)...x2x1 such lists, and when n gets at all large, one cannot answer questions about these lists by examining all of them, even with a very powerful computer. One type of problem of interest is, for given n and k, to figure out how many members of S(n) have exactly k positions i on the list where the number in position i is larger than i. The answer to this particular problem is well understood, but Shareshian and Michelle Wachs are working on various generalizations. With Axel Hultman, Svante Linusson and Jonas Sjostrand, Shareshian proved a conjecture of Alexander Postnikov relating a problem of the type described above to a problem involving slicing high dimensional spaces into pieces. He plans to work on generalizing this result. Finally, Shareshian is working towards proof a conjecture made by himself and Michael Aschbacher relating a class of algebraic objects (ubiquitous in pure and applied mathematics), called finite groups, to a class or combinatorial objects, called finite lattices. In all projects, it is hoped that both the solutions to the problems under consideration and the methods used to obtain them will be of use and interest to both combinatorialists and mathematicians working in other fields.