Scientific Visualization is one of the most active and exciting areas of Mathematics and Computing Science, and indeed one which is only beginning to mature. Current visualization algorithms break down for very large data sets. While present approaches use multi-resolution ideas, future data sizes will not be handled that way. New algorithms based on sophisticated mathematical modeling techniques must be devised which will permit the extraction of high-level topological structures that can be visualized. For these reasons a workshop was organized at the Banff International Research Station, focused specifically on mathematical issues. A primary objective of the workshop was to gather together a diverse set of researchers in the mathematical areas relevant to the recent advances in order to discuss the research challenges facing this field in the next several years. The workshop was organized into five different thrusts: > - Topology and Discrete Methods > - Signal and Geometry Processing > - Partial Differential Equations > - Data Approximation Techniques > - Massive Data Applications This book presents a summary of the research ideas presented at this workshop as well as a summary of the discussions.