Calculating the Inside View

The Geometrization Theorem of Thurston and Perelman provides a roadmap to understanding topology in dimension 3 via geometric means, by studying 3-manifolds which admit each of the eight Thurston Geometries. This talk concerns a recent project to produce provably correct intrinsic images and computer simulations of these geometric manifolds, by generalizing various algorithms from Euclidean geometry to locally homogeneous spaces. Doing so involves a careful analysis of the Riemannian-geometric features of each Thurston geometry, and we work out some new elementary properties of Nil, SL(2,R) and Sol. Joint work with Remi Coulon, Sabetta Matsumoto and Henry Segerman.