For some time, my research has centered on combinatorial properties of convex polytopes and, more generally, to algebraic approaches to combinatorial problems arising in geometry. Some questions are related to the facial structure of polytopes, for example, the enumeration of faces by dimension. Others have to do with subdivisions of polytopes.
More recently, I have been studying algebraic structures underlying the enumeration of faces and flags in polytopes and posets. This has led to the study of connections with the theory quasisymmetric and symmetric functions and has had application to enumeration in matroids and hyperplane arrangements and to a representation of the Kazhdan-Lusztig polynomials of Bruhat intervals in a Coxeter group.
- Quasisymmetric functions and Kazhdan-Lusztig polynomials (with F. Brenti), Israel Jour. Math. 184 (2011), 317–348.
- Peak quasisymmetric functions and Eulerian enumeration (with S. K. Hsiao and S. van Willigenburg), Advances in Mathematics 176 (2003), 248–276.
- Geometry of the space of phylogenetic trees (with S. Holmes and K. Vogtmann), Advances in Applied Mathematics 27 (2001), 733–767.
- Fiber polytopes (with B. Sturmfels), Annals of Math. 135 (1992), 527–549.
- Homology of smooth splines: generic triangulations and a conjecture of Strang, Trans. Amer. Math. Soc. 310 (1988), 325–340.
- Generalized Dehn-Sommerville relations for polytopes, spheres, and Eulerian partially ordered sets (with M. M. Bayer), Inv. Math. 79 (1985), 143–157.