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

TL;DR

This survey reviews recent algorithmic advances in computing topological invariants, especially Betti numbers, of semi-algebraic sets, highlighting key tools and open problems in the field.

Contribution

It summarizes recent progress in algorithms for topological invariants of semi-algebraic sets and discusses the main tools and open challenges.

Findings

Recent algorithms for Betti number computation

Integration of algebraic topology tools into semi-algebraic geometry

Identification of open problems in the field

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.