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:
Here's a cool example of a poset (Figure A). Its order complex can be obtained from the diagram in Figure B by an operation that is analogous to barycentric subdivision.
Figure A: A poset whose order complex is a torus. The order complex subdivides the cell complex shown in Figure B.
Poset diagram created with GAP and XGAP: source code.
Figure B: A cell complex for the torus.
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:
Figure C: L(S4), the subgroup lattice of the symmetric group on 4 letters.
L(S4) is an example of a comodernistic lattice.
Diagram created with GAP and XGAP.
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.