Research overviewMy research is broadly in geometric 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 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.
- 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.
Work of mine frequently involves the order complexes of lattices of subgroups or of cosets of a finite group G. Indeed, my thesis characterized the finite groups with a shellable or Cohen-Macaulay coset lattice. More recently, some highlighted work includes:
- In work with Jay Schweig, we've defined a new class of lattices (the comodernistic lattices) which seem to encapsulate well the combinatorial niceness of subgroup lattices of solvable groups. These lattices are CL-shellable, and we exhibited many examples of them. These lattices may be to solvable groups what supersolvable lattices are to supersolvable groups.
- In work with John Shareshian, we've shown that the lattice of cosets of any finite group is not contractible. We make thorough use of Smith Theory, along with the classification and certain generation theorems on finite simple groups.
Here's an example of a subgroup lattice, that of S4.
I'm also interested in independence complexes of graphs, and more broadly of clutters. Perhaps my strongest work here shows that the minimal non-shellable independence complexes of graphs (that is, flag complexes) are the independence complexes of cyclic graphics of lengths other than 3 or 5. This work answered a question of Michelle Wachs. The paper has been highly-cited, partially because it played an important role in popularizing the tool of vertex-decomposability for combinatorial commutative algebraists. Indeed, there is a strong connection to commutative algebra in this work, with the essential idea being that the (non-)face ideal of a simplicial complex Δ expresses Δ as an independence complex of some clutter.