This award supports the research of Professor Louis Billera of Cornell University. Dr. Billera's proposed work is in the areas of convex polytopes and multivariate splines, and will involve the use of algebraic and combinatorial methods to attack fundamental questions in each of these areas. Professor Billera's work is unusually interdisciplinary within mathematics, with roots in the applied field of spline functions as well as in the most abstract parts of commutative and homological algebra, algebraic geometry, and combinatorics. A spline is a thin strip of metal or wood that one may grasp at both ends and manipulate to form a smooth curve that not only passes through two points (the places where it is held), but does so in a given direction at each point. The spline functions of concern to Professor Billera generally describe smooth shapes of dimension greater than one, and with a smoother junction at the edges than can be gotten, in the case of physical splines, by holding the two strips together in the hand. They are of interest not only within mathematics, but in design, especially of automobiles and airplanes, and in solving problems in computer graphics.
