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.

- Lecture notes.
- Problem sheet.
- References
- Peter Scott, The geometries of 3-manifolds.
- Bruno Martelli, An Introduction to Geometric Topology.
- Greg Kuperberg, Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization.
- Culler, Dunfield, Goerner, Weeks, et. al. SnapPy, a computer program for studying the geometry and topology of 3-manifolds.

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 notes.
- Problem sheet.
- References
- Jeff Weeks, Computation of Hyperbolic Structures in Knot Theory
- Thurston's Lecture Notes, Chapters 3 and 4.
- Dunfield and Garoufalidis, Incompressibility criteria for spun-normal surfaces, Section 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 notes.
- Problem sheet.
- References
- Those from Lecture 2.
- Jeff Weeks,
*Convex hulls and isometries of cusped hyperbolic 3-manifolds.*Topology Appl.**52**(1993), no. 2, 127–149.

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.

- Lecture notes.
- Problem sheet.
- References
- Coulson, Goodman, Hodgson, and Neumann, Computing Arithmetic Invariants of 3-Manifolds.
- Hoffman, Ichihara, Kashiwagi, Masai, Oishi, and Takayasu, Verified computations for hyperbolic 3-manifolds.
- Dunfield, Hoffman, Licata, Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling.
- SageMath and CoCalc.