This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
This proposal is composed of five broad research topics in algebraic, geometric and topological combinatorics. The first project is to expand results about the strong Bruhat graph of Coxeter systems to the more general setting of balanced graphs. The PI has recently extended Billera and Brenti's work on the cd-index of Bruhat graphs to balanced graphs and showed the Kazhdan-Lusztig polynomials, an important topological and representation theoretic invariant, also extend to this setting. The second project is a study of the face incidence data of regular subdivisions of manifolds via the cd-index. The PI will investigate manifold analogues of classical results for convex polytopes, including to determine inequalities for the coefficients of the cd-index, to develop a theory of regular manifold arrangements in the spirit of Zaslavsky's seminal work on hyperplane arrangements, and to extend Stanley's notion of spherical shellability to manifolds. Motivated by Wachs' work on the d-divisible partition lattice, the third project is to examine the restricted partition lattice from the standpoint of poset homology, shellability and representation theory. The fourth project is a classical enumerative study of Dowling analogues of the Stirling numbers of the second kind and the Bell numbers. The fifth project is to investigate combinatorial permutation statistics, such as excedances, descents and the major index, for the group of affine permutations.
Combinatorics is inherently an interdisciplinary field of study linking many areas of mathematics and the sciences. This proposal further expands the range of combinatorics. For example, the Kazhdan-Lusztig polynomials are a deep invariant originally defined in topology. In this project they will be analyzed from a combinatorial perspective to further enhance our understanding of them. Theories that apply to polytopes, which are sphere-like objects, will be extended to other manifolds which have more complicated topological structure. Developing our topological perspective is important since a large part of modern-day physics is focused on studying the topology of space. Deepening our understanding of permutation statistics and basic combinatorial enumeration may help us to analyze and recognize patterns in vast genome data, as well as to improve communications, including the internet.
