Algebraic, topological and enumerative combinatorics

Shareshian将在枚举，代数和拓扑组合学的几个问题。 他将继续与M. Wachs的一个项目涉及准对称函数和排列统计。 他将努力扩大与A. Hultman，S. Linusson和J. Sjostrand涉及超平面排列和Bruhat序。 他将继续努力证明，有一些有限格L，使没有有限群G的子群格包含一个区间同构与L。 特别是，他将研究一类格出现在一个猜想，专门从事自己的拓扑猜想和组合猜想的M。大部分Shareshian的工作，这个项目在于交叉的三个领域的数学，即组合，拓扑和有限群理论。 重点是组合学，这是研究离散的，通常是有限的，赋予一些额外的结构集。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. 当n=3时，有3x 2x 1 =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）... x2 x1这样的列表，当n变得很大时，即使用非常强大的计算机，也不能通过检查所有这些列表来回答关于这些列表的问题。 一种有趣的问题是，对于给定的n和k，计算出S（n）中有多少个成员在列表上恰好有k个位置i，其中位置i的数字大于i。 这个特殊问题的答案是很好理解的，但Shareshian和Michelle Wachs正在研究各种概括。 与阿克塞尔Hultman，斯万特Linusson和乔纳斯Sjostrand，Shareshian证明了猜想的亚历山大Postnikov有关的问题的类型上述问题涉及切片高维空间成件。 他计划致力于推广这一结果。 最后，Shareshian正在努力证明一个猜想所作的自己和迈克尔Aschbacher有关一类代数对象（无处不在的纯数学和应用数学），称为有限群，一类或组合对象，称为有限格。 在所有的项目中，我们都希望所考虑的问题的解决方案和用于获得它们的方法将对其他领域的组合学家和数学家都有用和感兴趣。

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.