The geometrizat ion theorem of Thurston and Perelman provides a roadmap to understanding t opology in dimension 3 via geometric means. Specifically\, it states that every closed 3-manifold has a decomposition into geometric pieces\, and th e zoo of these geometric pieces is quite constrained: each is built from o ne of only eight homogeneous 3-dimensional Riemannian model spaces\, calle d the Thurston geometries. So to begin to understand what 3-manifolds "are like\," we may reduce the problem to first understanding these geometric pieces.
\nFor me\, the happy fact that our day-to-day life takes p lace in three dimensions is a major asset here\; while we can visualize su rfaces extrinsically\, and reason about 4-manifolds via slicing\, only for 3-manifolds can we really attempt to answer "what would it feel like/look like/be like" to live inside of one. To leverage our natural visual intui tion in three dimensions\, in joint work with Ré\;mi Coulon\, Sabett a Matsumoto\, and Henry Segerman\, we have adapted the computer graphics t echnique of raymarching to homogeneous Riemannian metrics. We use this to produce accurate and real-time intrinsic views of Riemannian 3-manifolds\; specifically the eight Thurston geometries and assorted compact quotients . In this talk\, I will take you on a tour of these spaces\, and talk a bi t about the mathematical challenges of actually implementing this.
\n LOCATION:https://researchseminars.org/talk/McGillGGT/24/ END:VEVENT END:VCALENDAR