Algebraic, topological and enumerative combinatorics

Shareshian将致力于枚举学、代数和拓扑组合学中的几个问题。他将继续与M·瓦克斯在一个涉及准对称函数和置换统计的项目上共同工作。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秩序。他将继续尝试证明存在有限格L，使得不存在子群格包含与L同构的区间的有限群G。特别地，他将考察出现在他自己的拓扑猜想和M.Aschbacher的组合猜想的猜想中的一类格。Shareshian在这个项目上的大部分工作位于三个数学领域的交集，即组合学、拓扑学和有限群论。重点是组合学，它是研究离散的，通常是有限的，被赋予了一些额外结构的集合。通常，组合学家希望知道有多少给定类型的结构存在，或者有多少这样的结构具有一些有趣的附加性质。例如，考虑数字1，…，n的所有可能列表的集合S(N)。当n=3时，有3x2x1=6这样的列表，即123,132,213,231,312,321。一般来说，这样的列表有x(n-1)x(n-2)…x2x1，当n变得很大时，即使用一台非常强大的计算机，也不能通过检查所有这些列表来回答有关这些列表的问题。一类有趣的问题是，对于给定的n和k，计算出S(N)中有多少成员在列表上恰好有k个位置i，其中位置i的数量大于i。这个问题的答案是众所周知的，但Shareshian和Michelle Waches正在进行各种推广。与Axel Hultman，Svante Linusson和Jonas Sjostrand一起，Shareshian证明了Alexander Postnikov的猜想，将上述类型的问题与涉及将高维空间分割为碎片的问题联系起来。他计划致力于推广这一结果。最后，Shareshian正在努力证明他和Michael Aschbacher提出的一个猜想，该猜想将一类称为有限群的代数对象(在纯数学和应用数学中普遍存在)与一类或组合对象称为有限格。在所有项目中，希望正在考虑的问题的解决方案和获得它们的方法都将对从事其他领域工作的组合学家和数学家有用和感兴趣。

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.