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 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.

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 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.

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 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.

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.

• 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.