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 3-manifold has a decomposition into geometric pieces, and the zoo of these geometric pieces is quite constrained: each is built from one of eight homogeneous 3-dimensional Riemannian model spaces (called the Thurston geometries).

In this talk, we will approach the question of “what does a 3-manifold 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 re-assemble these geometric pieces and understand an “inside view” of general 3-manifolds.

### Occurrences

I’ve given a version of this talk to several audiences, often at the math-department-colloquium 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.