Projects and Grants per year
Grants and Contracts Details
Description
The PI's most signicant recent research touches on three themes: (i) face incidence
enumeration for geometric objects, (ii) a new theory of qanalogues 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 nontrivial intervals satisfy the
EulerPoincare 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 quasigraded 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 cdindex, a noncommutative polynomial which
removes all linear redundancies among the face incidence data. Questions include finding a
natural interpretation of the coefficients of the cdindex which, unlike the case of polytopes,
can now be negative, ending inequalities for the associated cdindex of Whitney stratifed
spaces, and extending algebraic structures involved with traditional flag enumeration to
the stratifed arena.
Motivated by recent work of FuReinerStantonThiem on the qbinomial, the PI and
the PI's graduate student Cai are pursuing the larger question of understanding classical
qanalogues which can be written more compactly in terms of powers of q and 1 + q. For
qStirling numbers they have discovered a compact encoding of the original qanalogue data
which reveals new poset theoretic, topological and homological insight for qanalogues.
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 workinprogress 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
Status  Finished 

Effective start/end date  9/1/16 → 8/31/22 
Funding
 Simons Foundation: $35,000.00
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Projects
 1 Finished

Investigations in Combinatorial Mathematics  Department Funds
9/1/16 → 8/31/22
Project: Research project