Math 595 AT3, Algorithmic topology and geometry of 3-manifolds: theory and practice

Fall 2021

Course Description

In dimensions four and higher, most basic questions about manifolds (e.g. is a given manifold the n-sphere?) are algorithmically undecidable. In contrast, many questions about 3-manifolds are not just decidable but have practical algorithms that have been implemented and run on literal millions of 3-manifolds. I will survey some of what is known here, focusing on the use of geometry to solve topological problems in the spirit of Thurston.

Topics will include basics of 3-dimensional topology and the Geometrization Theorem, solvability of the word and homeomorphism problems for 3-manifolds, and verified computation using interval arithmetic to compute hyperbolic structures on 3-manifolds. The exact mix of topics will depend on students' background and interests, but to get the general flavor, see the notes, references, and handouts from a summer school course I taught in 2017. The course is independent from my other 595 course this term and I will minimize the overlap with Eric Sampterton's course from Spring 2021.

Prerequisites: Basic knowledge of smooth manifolds and algebraic topology, e.g. Math 518 and Math 525. No prior knowledge of 3-manifolds will be assumed, but at least a vague interest in computation is recommended.


Students registered for the course will need to write a short (2-4 page) paper which will be due on Friday, December 10th. This paper is largely free-form, and can be about any subject related to the content of this course. For instance, it could be a brief account of a result not covered in class, a review of the some related results explaining why they are interesting, a detailed work-out of a proof only sketched in class, or careful solutions to problems from class or taken from our various readings. Alternatively, code and/or computations can be substituted for the paper. One source of topics would be these problem sheets.

Lecture notes

Notes from each lecture will be posted here.

  1. Oct 18. Introduction. Some references:
  2. Oct 20. Basic examples. References for basic hyperbolic geometry include Scott and Martelli above as well as
  3. Oct 22. Geometry of Cusps. Did 1-4.
  4. Oct 25. From triangulations to hyperbolic structures. Did 1-4.
  5. Oct 27. Thurston's gluing equations. Did 1-4.
  6. Oct 29. Hyperboloid model. Did 1-4.
  7. Nov 1. Canonical cell decompositions. Did 1-4.
  8. Nov 3. More on canonical cell decompositions. Did 1 to middle of 4.
  9. Nov 5. Finding the canonical decomposition.
  10. Nov 8. Canonical decompositions in 3D; closed hyperbolic manifolds.
  11. Nov 10. More on closed manifolds.
  12. Nov 12. Hyperbolic Dehn filling. Through example on page 4.
  13. Nov 15. More on Hyperbolic Dehn filling. Did 1-4.
  14. Nov 17. Volumes of hyperbolic 3-manifolds. Did everything.
  15. Nov 19. Office hour instead of class. Come by my office if you want topic ideas or references for your final paper, or want to discuss the material so far.
  16. Nov 29. Certifying solutions to gluing equations.
  17. Dec 1. The HIKMOT method.
  18. Dec 3. Applications of verified hyperbolic structures.
  19. Dec 5. Proof by parameter space.
  20. Dec 7. Machine learning for fun and profit.

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