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# Louis J. Billera

Associate Chair for Language Instruction

## Overview

### Research Focus

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

### Departments/Programs

- Mathematics

### Graduate Fields

- Applied Mathematics
- Mathematics
- Operations Research and Information Engineering

## Publications

*Generalized Dehn-Sommerville relations for polytopes, spheres, and Eulerian partially ordered sets* (with M. M. Bayer), Inv. Math. **79** (1985), 143–157.

*Homology of smooth splines: generic triangulations and a conjecture of Strang,* Trans. Amer. Math. Soc. **310** (1988), 325–340.

*Fiber polytopes* (with B. Sturmfels), Annals of Math. **135** (1992), 527–549.

*Geometry of the space of phylogenetic trees *(with S. Holmes and K. Vogtmann), Advances in Applied Mathematics **27** (2001), 733–767.

*Peak quasisymmetric functions and Eulerian enumeration* (with S. K. Hsiao and S. van Willigenburg), Advances in Mathematics **176** (2003), 248–276.