Topology of random simplicial complexes: a survey

This expository article is based on a lecture from the Stanford Symposium on Algebraic Topology: Application and New Directions, held in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madsen.Comment: After revisions, now 21 pages, 5 figure

A Robustness Study for the Extraction of Watertight Volumetric Models from Boundary Representation Data

Geometrically induced topology plays a major role in applications such as simulations, navigation, spatial or spatio-temporal analysis and many more. This article computes geometrically induced topology useful for such applications and extends previous results by presenting the unpublished used algorithms to find inner disjoint (d+1)-dimensional simplicial complexes from a set of intersecting d-dimensional simplicial complexes which partly shape their B-Reps (Boundary Representations). CityGML has been chosen as the input data format for evaluation purposes. In this case, the input data consist of planar segment complexes whose triangulated polygons serve as the set of input triangle complexes for the computation of the tetrahedral model. The creation of the volumetric model and the computation of its geometrically induced topology is partly parallelized by decomposing the input data into smaller pices. A robustness analysis of the implementations is given by varying the angular precision and the positional precision of the epsilon heuristic inaccuracy model. The results are analysed spatially and topologically, summarised and presented. It turns out that one can extract most, but not all, volumes and that the numerical issues of computational geometry produce failures as well as a variety of outcomes

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

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.Comment: 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 Ed