Steve Trettel

Abstract
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 3manifold has a decomposition into geometric pieces, and the zoo of these geometric pieces is quite constrained: each is built from one of eight homogeneous 3dimensional Riemannian model spaces (called the Thurston geometries).
In this talk, we will approach the question of “what does a 3manifold look like” from the perspective of geometrization. Through animations of simple examples in dimensions 2 and 3 we review what it means to put a (complete, homogeneous) geometric structure on a manifold, and construct an example admitting each of the Thurston geometries.
Using software written in collaboration with Remi Coulon, Sabetta Matsumoto and Henry Segerman, we will explore these manifolds ``from the inside''  that is, simulating the view one would have in such a space by raytracing along geodesics. Finally we will touch on how to reassemble these geometric pieces and understand an “inside view” of general 3manifolds.
Occurrences
I’ve given a version of this talk to several audiences, often at the mathdepartmentcolloquium level. Here’s a short selection:
 Cornell University
 University of Florida
 McGill University
 Gathering for Gardner, Celebration of Mind
 University of Arkansas
Here’s a recording of the talk as given virtually at Cornell, during the Cornell Topology Festival.
And here’s a presentation aimed at a more general audience on similar material, given for Gathering for Gardner, online.
Slides
Here are slides from a recent version of this talk!