Splines on Triangulations of Arbitrary Topology

There is a well-developed theory of bivariate splines (or piecewise polynomials) on planar triangulations, which allow us, in principle, to construct splines of arbitrary smoothness. However, such splines can only be used to obtain parametric surfaces that are topologically equivalent to a planar region. This theory of bivariate splines fails completely when one wants to define smooth surfaces of arbitrary topological genus. In this talk we shall address the causes of this serious gap in the spline theory and discuss possible remedies. This is joint work with Carolina Beccari (Univ. of Bologna) and Dan Gonsor (The Boeing Co.).