# Louis Billera

Ph.D. (1968) City University of New York

### Research Area

Geometric and algebraic combinatorics

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.

A common theme in much of this has been the construction of polytopes to given specifications: for example the construction with Carl Lee of polytopes satisfying the conditions of McMullen’s g-conjecture, showing these conditions to be sufficient to describe the enumeration of faces of all simplicial convex polytopes, or the construction with Bernd Sturmfels of fiber polytopes, showing that certain sets of polyhedral subdivisions of polytopes themselves have the structure of convex polytopes. In addition, we have used some of these ideas in the study of questions arising in biology concerning the structure of the space of all phylogenetic trees.

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 represention of the Kazhdan-Lusztig polynomials of Bruhat intervals in a Coxeter group.### Selected Publications

*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.