Research interests

Quick links:
     CV (as of January 19, 2012)
     Published papers
     Preprints
     Google Scholar profile

A narrative overview:
My research is broadly in geometric combinatorics, including connections with group theory and commutative algebra.

Geometric combinatorics starts with a combinatorial object, such as a graph or partially ordered set (poset), and attaches a topological space, usually a simplicial complex. If you recall that a simplicial complex is a set system closed under inclusion, this appears very natural. For example:

In a graph G: any subset of an independent set is independent. Thus, the set of all independent sets of G is a simplicial complex, called the independence complex.
In a poset P: any subset of a chain (totally ordered subset) is also a chain. Thus, the set of all chains of P is a simplicial complex, called the order complex.

Often, one can then relate the topology of this simplicial complex back to the combinatorics of the object one started with.

           Face lattice of a torus            Cell complex for a torus
               A poset whose order complex is a torus.
               (The order complex subdivides the pictured cell complex.)
               Poset diagram created with GAP and XGAP: source code.

1. Subgroup and coset lattices. For example, the subgroup lattice of a finite group is the poset consisting of all subgroups, ordered by inclusion. Shareshian showed that a finite group is solvable if and only if its subgroup lattice is shellable. (Here, shellable is a combinatorial condition which essentially says that the facets / maximal chains "fit nicely together".) Shellability seems to be a useful condition for recognizing nice classes of groups from their lattices. Some related results:

In my thesis, I showed that the coset lattice of a finite group G is shellable if and only if G is complemented.
I've constructed an EL-shelling of the subgroup lattice of a finite solvable group. (This simultaneously shows the subgroup lattice is shellable and describes the homotopy type.)
I've given a new topological characterization of solvable finite groups, in terms of the depth of their subgroup lattices. The group theory aspect of this follows from joint work with John Shareshian in which we related the chief series of the group, other modular chains in the subgroup lattice, and the minimum length of a maximal chain in the subgroup lattice.

L(S₄), the subgroup lattice of the symmetric group on 4 letters.
Diagram created with GAP and XGAP.

2. Independence complexes and commutative algebra. Another area I'm interested in is the geometric combinatorics of independence complexes of graphs and clutters. These are very general objects, and a general characterization of (for example) which such complexes are shellable would be unexpected. However, there are occasionally interesting connections between pure graph theory/combinatorics and the geometry of an independence complex.

There is also a connection to commutative algebra, via the face ring or, equivalently, via the edge ideal. The simplest version of this is as follows: given a graph G with vertex set [n], take a polynomial ring with n generators, and consider the ideal generated by x_i x_j for {i,j} ranging over the edge set. In this context shellability (and the closely-related Cohen-Macaulay property) are of great interest to commutative algebraists.

The interplay between these three fields (graph theory, geometric combinatorics, and commutative algebra) can lead to progress. One example of this interplay is as follows:
A chordal graph is a graph with no induced cycles of length longer than 3. It was proved by commutative algebraists Van Tuyl and Villarreal (motivated by more purely algebraic work of Francisco and Van Tuyl) that the independence complex of a chordal graph is shellable. I extended this to show that a graph with no induced cycles of length other than 3 or 5 is shellable. The proof applies a graph theoretic lemma of Chvátal, Rusu, and Sritharan to find a geometric decomposition giving a shelling. Since the independence complexes of other cyclic graphs are not shellable, this result characterizes the obstructions to shellability (minimal non-shellable complexes) that are independence complexes of graphs, answering a question of Wachs.

3. Combinatorial consequences. One can prove purely combinatorial theorems about simplicial complexes (and related posets / graphs) obeying regularity conditions arising from geometric considerations. One result of this nature that I've proved is an Erdős-Ko-Rado type intersection theorem for faces of a shellable cone. (This answers a special case of a conjecture of Holroyd, Talbot, and Borg.) Another such result is joint work with Stephan Foldes, where we've shown that the antichain cutsets of a ranked poset which is sufficiently connected consist exactly of the level sets.

More information:
My preprints (with abstracts) are the definitive summary of what I've been working on lately. My published papers are also available, as is my curriculum vitae (version of Jan 19, 2012).

I also have some mathematical software, including my Mac OS X front-end Gap.app for the GAP computer algebra system.