Lectures by Nathan Dunfield at the University of Warwick during September 11-15, 2017.
The role of geometry in 3-dimensional topology, the Geometrization Theorem, Mostow rigidity and how this solves the homeomorphsim in dimension 3. Show this is all actually practical via a basic tutorial on SnapPy.
Complete hyperbolic manifolds of finite volume. The case of surfaces: topological ideal triangulations, shear coordinates, and incompleteness phenomena. Ideal triangulations of 3-manifolds with the example of mapping tori. Geometric ideal tetrahedra in hyperbolic 3-space. Edge equations and the deformation variety with connections to character varieties.
Finding hyperbolic structures by solving Thurston's gluing equations. Application of canonical cell decompositions to solving the homeomorphism problem for hyperbolic 3-manifolds.
Proving a manifold is hyperbolic by rigorously certifying a solution to Thurston's gluing equations. Two approaches: arithmetic geometry and interval analysis. Demonstration of SnapPy in SageMath.