The proposed research is on the combinatorial and algebraic theory of smooth piecewise polynomial functions over polyhedral subdivisions and its relationship to the theory of facial enumeration in convex polyhedra. A particular emphasis will be given to the problem of determining a simple generating set for the algebra of all smooth splines, as was already done in the case of continuous piecewise polynomials, and to the problem of determining for which subdivisions does this algebra have a free basis over the ring of global polynomial functions. The PI has shown these problems to be related to certain enumeration questions for convex polytopes, and he hopes to develop this connection further here. Piecewise polynomial functions have long been used for solving partial differential equations by the finite element method: this work was originally motivated by questions arising in this context. More recently, they have found extensive application in the area of surface modelling for computerized design and control. As a result of earlier work on this project, they are now beginning to find new applications in the theory of structural rigidity.
