UC San Diego

One of the principle concerns of computational mathematics is the discrete representation and approximation of mathematical objects. It is common for classical definitions of mathematical objects to allow for elegant mathematical analysis yet lead to computational models which are either inefficient or unimplementable. One mathematical object that fits this mold is a differentiable manifold. Differentiable manifolds are of increasing interest in modern computational mathematics as more geometrically complex problems are considered. In this dissertation, we propose a computational model for representing compact $C^1$ differentiable manifolds without boundary and their function spaces. This model is based on a combination of simplicial complexes and splines. Simplicial complexes are a standard tool for computing in both the pure and applied math settings. Splines are piecewise polynomials relative to some tessellation of a domain of interest whose coefficients have been chosen to enforce differentiability at all points of the domain.