Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems

Abstract

We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semi-algebraic sets. Aside from describing these results, we discuss briefly the background as well as the importance of these problems, and also describe the main tools from algorithmic semi-algebraic geometry, as well as algebraic topology, which make these advances possible. We end with a list of open problems.

Survey article, 74 pages, 15 figures. Final revision. This version will appear in the AMS Contemporary Math. Series: Proceedings of the Summer Research Conference on Discrete and Computational Geometry, Snowbird, Utah (June, 2006). J.E. Goodman, J. Pach, R. Pollack Eds