Our research on graphics and discrete differential modeling is based around the insight that numerical simulation and modeling on a computer require discrete versions of the continuous mathematical models describing these systems. For these computations to be truly predictive, reliable, and efficient the underlying continuous structures must be transferred to the discrete systems. The goal of the research at Caltech in this area is to identify the relevant mathematical and computer science tools to lay the foundation for the rational construction of such discrete differential modeling tools which preserve the relevant structures. The study of geometry in a broad sense forms the core of this area but it also draws considerably on fields ranging from algebraic topology to computational geometry, graph theory, combinatorics, applied mathematics, and computer science. Application areas include computer graphics, variational mechanics, and biological systems.