Algebraic, topological and enumerative combinatorics

Shareshian将研究枚举、代数和拓扑组合学中的几个问题。他将继续与M. Wachs在一个涉及拟对称函数和置换统计的项目上的合作。他将尝试扩展与A. Hultman， S. Linusson和J. Sjostrand的联合工作，涉及超平面排列和Bruhat秩序。他将继续尝试证明存在一个有限格L，使得有限群G的子群格不包含与L同构的区间。特别是，他将检验出现在一个猜想中的一类格，这个猜想既专门研究他自己的拓扑猜想，也专门研究M. Aschbacher的组合猜想。Shareshian在这个项目上的大部分工作涉及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的，通常是有限的集合的研究。通常，组合学家希望知道给定类型存在多少结构，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个这样的列表，即123,132,213,231,312,321。一般来说，有x(n-1)x（n-2）…X2x1个这样的列表，当n非常大的时候，我们不能通过检查所有的列表来回答关于这些列表的问题，即使有一台非常强大的计算机。我们感兴趣的一类问题是，对于给定的n和k，计算出S(n)中有多少个元素恰好在列表的第k个位置i上，而第i个位置上的数字大于i。这个问题的答案很容易理解，但Shareshian和Michelle Wachs正在研究各种推广方法。与Axel Hultman， Svante Linusson和Jonas Sjostrand一起，Shareshian证明了Alexander Postnikov的一个猜想，该猜想将上述类型的问题与涉及将高维空间切成块的问题联系起来。他计划推广这一结果。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）与一类称为有限格的组合对象联系起来。在所有的项目中，我们都希望所考虑的问题的解和得到这些解的方法对在其他领域工作的组合学家和数学家都有用处和兴趣。

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.