Algebraic, topological and enumerative combinatorics

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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.