Visualizing Riemannian Geometry for Research and Outreach
The Geometrization Theorem of Thurston and Perelman provides a roadmap to understanding topology in dimension 3 via geometric means. Specifically, it states that every closed 3-manifold has a decomposition into geometric pieces, and each piece is realizable as a finite volume quotient of one of eight homogeneous spaces (the Thurston geometries). I will talk about a project (joint work with Remi Coulon, Sabetta Matsumoto and Henry Segerman) to produce accurate computer graphics simulations of 3-manifolds with these geometries. I'll point out some of the mathematical and computational challenges to doing this, and some of the outreach activities that portions of the project have been repurposed for.