General Biographic Information - L.J. Billera

B.S. 1964 (Rensselaer Polytechnic Institute); M.A. 1967, Ph.D. 1968 (City University of New York)


Billera joined the Cornell faculty in 1968, after having been an Educational Testing Service psychometric fellow at Princeton University in 1964-65 and an NDEA Graduate Fellow at the City University of New York. He was a National Science Foundation postdoctoral fellow at the Hebrew University of Jerusalem and at Cornell in 1969. In 1974-75 he was a visiting research associate in the Department of Mathematics at Brandeis University, and in 1980 he was a Professeur Invité at the Center for Operations Research and Econometrics, Université Catholique de Louvain in Belgium. He served as the first associate director of the National Science Foundation Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) at Rutgers University in 1989. During the Spring terms of 1992 and 2005, he visited the Mittag-Leffler Institute outside Stockholm, and during 1996-7, he was a Research Professor at the Mathematical Science Research Institute in Berkeley.

Billera is a member of the American Mathematical Society and the Mathematical Association of America. In 1994, he was awarded the Fulkerson Prize in Discrete Mathematics by the American Mathematical Society and the Mathematical Programming Society for his research in multivariate splines. In 2010, he gave an invited lecture on his work at the International Congress of Mathematicians in Hyderabad, India.

Research Interests

``My research is concerned with the application of algebraic techniques to combinatorial problems, most often those arising in discrete and convex geometry. One problem which has attracted the interest of mathematicians for hundreds of years is that of enumerating the faces of convex polytopes. Great progress has been made on this problem in recent years, mainly as a result of the introduction of techniques from commutative algebra and algebraic geometry. My interest in this area continues. Most recently, we have applied techniques from this area to the study of Coxeter groups.

`` The methods developed to deal with this geometric enumeration problem have been applied to the study of piecewise polynomial functions on simplicial complexes. These functions, also known as finite elements or multivariate splines, are of considerable interest in methods for solving partial differential equations and for generating surfaces for computer-graphic display and machine control. The methods of algebraic combinatorics have provided a great deal of insight into this class of functions. Most notably, they have allowed the introduction of symbolic computation methods to this area.

`` Related to this is the study of the properties of the set of all subdivisions of a given region using a prescribed collection of vertices. This has been shown, under general conditions, to have the structure of a convex polytope. Study of this polytope and its applications is another focus of my research. One outgrowth of this work has been our discovery of unexpected complications in the structure of the polytope underlying the classical Traveling Salesman problem of combinatorial optimization.'' 

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