Applied Topology and Algorithmic Semi-Algebraic Geometry

Applied topology is a rapidly growing discipline aiming at using ideas coming from algebraic topology to solve problems in the real world, including analyzing point cloud data, shape analysis, etc. Semi-algebraic geometry deals with studying properties of semi-algebraic sets that are subsets of Rnand defined in terms of polynomial inequalities. Semi-algebraic sets are ubiquitous in applications in areas such as modeling, motion planning, etc. Developing efficient algorithms for computing topological invariants of semi-algebraic sets is a rich and well-developed field. However, applied topology has thrown up new invariants—such as persistent homology and barcodes—which give us new ways of looking at the topology of semi-algebraic sets. In this thesis, we investigate the interplay between these two areas. We aim to develop new efficient algorithms for computing topological invariants of semialgebraic sets, such as persistent homology, and to develop new mathematical tools to make such algorithms possible