Steve Trettel (Stanford University)
Speaker: Steve Trettel (Stanford University)
Title: Raymarching the Thurston geometries: visual intuition for geometric topology in 3 dimensions.
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 only eight homogeneous 3-dimensional Riemannian model spaces, called the Thurston geometries. So to begin to understand what 3-manifolds "are like," we may reduce the problem to first understanding these geometric pieces. For me, the happy fact that our day-to-day life takes place in three dimensions is a major asset here; while we can visualize surfaces 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 intuition in three dimensions, in joint work with R茅mi Coulon, Sabetta Matsumoto, and Henry Segerman, we have adapted the computer graphics technique 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 bit about the mathematical challenges of actually implementing this.
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Link:
Zoom ID: 989 1072 6246
Password: delta
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