Abstract:

This talk is an exploration of W. P. Thurston’s beautiful paper ‘Shapes of Polyhedra and Triangulations of the Sphere’, which examines moduli spaces of polyhedra arising from triangulations of the sphere. We will seek to understand the first result of the paper (namely, that these spaces are locally isometric to a complex hyperbolic space) using Alexandrov’s work on convex polyhedra. I will define all required basic definitions in a friendly and intuitive manner and use these to build the proof in real-time using a slightly different, more combinatorial approach to the material based on lecture notes by R. Schwartz. The talk is largely self-contained; there will be many enjoyable pictures and, if time permits, some pleasant geometric digressions.