LMS-CMI Research School: Computation of geometric structures

Lectures by Nathan Dunfield at the University of Warwick during September 11-15, 2017.

Lecture 1

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.

Lecture 2

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.

Lecture 3

Finding hyperbolic structures by solving Thurston's gluing equations. Application of canonical cell decompositions to solving the homeomorphism problem for hyperbolic 3-manifolds.

Lecture 4

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.

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