Non-Linear Computational Geometry

One of the original goals in the historical development of algebraic geometry was to understand the behavior of curves and surfaces in three dimensions. The recent theoretical and technological advances in areas such as Robotics, Manufacturing, Computer Vision and Computer-Aided Geometric Design, in synergy with the increased availability of computational resources, have brought this aspect once more into the forefront of research.

Mechanical design problems (like the relatively simple device known as the Stewart platform), or questions such as the determination of all aspects graphs in computer vision or the implicitation of surfaces, can prove extremely challenging from the computational algebraic geometry perspective. The need for reliable algorithms for these and other problems has provided stimulus for an exciting new array of closely interrelated techniques, that include both symbolic and numerical elements, and that are increasingly demonstrating their practical relevance. Applications of algebraic geometry in these areas are expected to interact well with the proven techniques of piecewise-linear computational geometry (e.g. Voronoi diagrams, hyperplane arrangements).

This workshop will explore the applications of algebraic tools in computational geometry. Tools from Elimination Theory and Resultants will play an important role, as will effective methods in real algebraic geometry (e.g., the work of Emiris, Gonzalez-Vega, Manocha, Mourrain, Roullier, Sottile, etc.). The results have encouraged research directions emphasizing the complexity-theoretic aspects (e.g., the work of Basu), but also have enabled many successful applications in computer vision and geometric design (e.g. Petitjean, Pottmann, Sederberg, etc.)