Investigations in Combinatorial Mathematics - Department Funds

The PI's most signicant recent research touches on three themes: (i) face incidence enumeration for geometric objects, (ii) a new theory of q-analogues and (iii) topological number theory. One of the main goals of the PI's research has been to study the face incidence struc- ture of polytopes, and more generally, the chain enumerative data of an Eulerian partially ordered set (poset). Recall a poset is Eulerian if all of its non-trivial intervals satisfy the Euler-Poincare relation. The most recent development is the PI's work with Ehrenborg and Goresky on topological flag enumeration for Whitney stratifed spaces and the larger class of Eulerian quasi-graded posets. These results have widened the traditional study of face incidences of polytopes by expanding the topological objects being studied and the nature of the questions being asked regarding the cd-index, a noncommutative polynomial which removes all linear redundancies among the face incidence data. Questions include finding a natural interpretation of the coefficients of the cd-index which, unlike the case of polytopes, can now be negative, ending inequalities for the associated cd-index of Whitney stratifed spaces, and extending algebraic structures involved with traditional flag enumeration to the stratifed arena. Motivated by recent work of Fu-Reiner-Stanton-Thiem on the q-binomial, the PI and the PI's graduate student Cai are pursuing the larger question of understanding classical q-analogues which can be written more compactly in terms of powers of q and 1 + q. For q-Stirling numbers they have discovered a compact encoding of the original q-analogue data which reveals new poset theoretic, topological and homological insight for q-analogues. Future research avenues include connections with symmetric function theory, the cyclic sieving phenomenon and interpolating statistics. An area of research on the horizon is the PI's joint work-in-progress with Ehrenborg, Govindaiah and Park to understand the topology of the van der Waerden complex. This is a simplicial complex related to arithmetic progressions of length k in the set {1,..., n}. Bounds have been established for the dimension of the homotopy type as well as for when the complex is contractible. It remains to sharpen the study of the homotopy type, the Betti numbers and possible connections to number theory. 1