☰Research interests

Research overview

My research is broadly in geometric and topological combinatorics, including its connections with group theory and commutative algebra.

Geometric combinatorics uses geometric and/or topological techniques to study combinatorial problems. The problems that I am interested in usually begin with a combinatorial object, such as a graph or partially ordered set, and attach a simplicial complex. Since a simplicial complex is just a set system that is closed under inclusion, they arise naturally in combinatorics. 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. A simplicial complex that can be represented as the independence complex of a graph is called a flag 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 of P. The Euler characteristic of P is useful in solving inclusion-exclusion problems (via Möbius inversion).

Here's a cool example of a poset (Figure A). Its order complex is a torus, and can be obtained from the cell complex in Figure B by an operation that is analogous to barycentric subdivision.

I'm frequently interested in several related properties: of Cohen-Macaulay, a topological property; of vertex-decomposability and/or EL-shellability, properties asserting combinatorial decompositions; and of shellability, a property intermediate between combinatorics and topology. I particularly like it when I can use a tool from one area of math to settle a question in another area.

Much like history, my work does not repeat itself, but it sometimes rhymes. Threads that have reoccurred throughout my work include combinatorial commutative algebra, topological aspects of lattices of subgroups and/or cosets, and generalizations of set intersection theorems.

Combinatorial commutative algebra

There is a natural link between commutative algebra and independence complexes. Indeed, the nonface ideal of a simplicial complex is generated by monomials supported on nonfaces. Thus, the nonface ideal of Δ expresses Δ as the independence complex of some graph or hypergraph. If the simplicial complex is presented as an independence complex, then it is particularly easy to read off the associated ideal. Topological and combinatorial conditions often translate naturally to algebraic conditions.

My most-cited result characterizes the minimal non-shellable flag complexes: they are exactly the independence complexes of cyclic graphs of length other than 3 or 5. This work answered a question of Michelle Wachs. The paper helped popularize the tool of vertex-decomposability among combinatorial commutative algebraists. The cyclic graphs of length other than 3 are the obstructions to chordality of the graph, and other work of mine constructs a candidate family of clutters (simple hypergraphs) that one could call chordal. The independence complex of such a chordal clutter is also shellable and so sequentially Cohen-Macaulay.

Some other work in this area is on the Castelnuovo-Mumford regularity of a square-free monomial ideal. Here, a bound of Kalai and Meshulam translates very nicely into graph theory: the regularity of the ideal associated to a graph is bounded from above by a graph-theoretic invariant, the cochordal cover number.

Lattices of subgroups, lattices of cosets

Work of mine frequently involves the order complexes of lattices of subgroups or of cosets of a finite group G. Topology of these complexes often lets one detect natural group-theoretic properties, such as (super)solvability or complementedness.

Here's an example of a subgroup lattice, that of the symmetric group S₄.

The subgroup lattice of a group has been studied since the earliest days of lattice theory. The topological point of view appears to be helpful in this setting. Indeed, Shareshian showed in 2001 that a group is solvable if and only if the order complex of its subgroup lattice is shellable. In older work, I gave a natural EL-shelling of the subgroup lattice of a solvable group. More recently, with Jay Schweig we defined the class of comodernistic lattices, which encapsulate much of the combinatorial niceness of subgroup lattices of solvable groups. We showed these lattices to be CL-shellable (later improved by Tiansi Li to an EL-shelling); many of the known classes of shellable lattices lie in this class. Comodernistic lattices may be to solvable groups as supersolvable lattices are to supersolvable groups.

The topological point of view is also helpful for studying the coset lattice. Indeed, classic definitions of lattice theory tend to be uninteresting here, while the topological view has something to say.

My thesis characterized the finite groups with a shellable or Cohen-Macaulay coset lattice. Later, in work with John Shareshian, we proved that there is no finite group whose coset lattice is contractible. Indeed, Bob Guralnick, Shareshian and I recently showed that the coset lattice of any finite group has nontrivial rational homology.

Proofs of non-vanishing homology of a coset lattice make thorough use of Smith-Oliver theory together with new (and ongoing) work on invariable generation of finite simple groups.

Generalized set intersection theorems

The Erdős-Ko-Rado theorem says that the largest possible pairwise-intersecting uniform family of small subsets of a base set is obtained by taking all the subsets containing a fixed element. I've considered several extensions of this result.

In older work, I showed that an analogue of the Erdős-Ko-Rado bound holds for families of faces in a cone over a Cohen-Macaulay simplicial complex (and somewhat more general complexes). In recent work with Denys Bulavka, we extended this to a strict (structural) result.

Another direction that I have recently been interested in extends the setting of an intersecting family of sets to that of a self-annihilating subspace of an exterior algebra. There is an attractive relationship with algebraic groups and with broader algebraic geometry. Indeed, the combinatorial shifting invented by Erdős, Ko, and Rado may be viewed as the limit of an action of a curve formed by transvection matrices. Part of the work on this is joint with Bulavka and with Francesca Gandini.

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 June 10, 2026).

I also have some mathematical software, notably including Gap.app, my macOS front-end and distribution for the GAP computer algebra system. Computer experiments are frequently a starting point for my work.

If you are a student who is interested in working on a project with me, then you might be interested in seeing some past projects on my advising page.