This award supports the interdisciplinary research in piecewise polynomial approximation, combinatorics, and algebra of Professor Louis Billera of Rutgers University. Dr. Billera's work will involve the use of algebraic and combinatorial methods to attack fundamental questions in the two areas of multivariate splines and convex polytopes. His previous work has shown that the methods of algebraic combinatorics developed for the study of enumeration questions for convex polytopes are also useful in the study of dimension questions for multivariate splines. The present research will build on these previous discoveries. A spline is a thin strip of wood or metal that one may manipulate from the ends, to produce a smooth curve. In the simplest (one-dimensional) case, a mathematical spline is a curved graph segment for which one may specify the way in which the endpoints are held, to produce a smooth curve. Multivariate splines belong to functions of several variables in the way that the splines mentioned above belong to functions of one variable. They furnish excellent and efficient tools for drawing or describing curved surfaces, and are of great importance not only in theoretical applied mathematics, but also in such fast- developing areas as computer-aided design.
