Steve Trettel



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 (to our knowledge) new elementary properties of Nil, SL2R and Sol. I will mainly speak on the interesting mathematics involved, but there will be many pretty pictures along the way! All is joint work with Remi Coulon, Sabetta Matsumoto and Henry Segerman.


I’ve given several versions of this talk in different settings (to grad students, to other researchers, in colloquia). Some of the recent ones are listed below:

  • Arizona State University, 2020
  • Stanford University, 2020
  • Weizmann Institute, 2021
  • UT Austin, 2022


Here are slides from a recent version of this talk!