**Time and Place:**MWF from 11:00-11:50am in 143 Altgeld Hall. Second half of semester only, starts Oct 18.**Section:**AT3**CRN:**57742**Instructor:**Nathan Dunfield**E-mail:**nmd@illinois.edu**Office:**378 Altgeld.**Office Phone:**(217) 244-3892**Office Hours:**Mon and Wed at 3pm; other times by appointment.

**Web page:**http://dunfield.info/595B**Lecture notes**

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.

Notes from each lecture will be posted here.

- Oct 18. Introduction. Some 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.

- Oct 20. Basic examples.
References for basic hyperbolic geometry include Scott and Martelli
above as well as
- Bonahon, Low-Dimensional Geometry (the most elementary of all these sources).
- Thurston, Three-Dimensional Geometry and Topology.

- Oct 22. Geometry of Cusps. Did 1-4.
- Oct 25. From triangulations to
hyperbolic structures. Did 1-4.
- Jeff Weeks, Computation of Hyperbolic Structures in Knot Theory.

- Oct 27. Thurston's gluing equations. Did 1-4.
- Oct 29. Hyperboloid model. Did 1-4.
- Nov 1. Canonical cell
decompositions. Did 1-4.
- Jeff Weeks,
Convex hulls and isometries of cusped hyperbolic 3-manifolds.
Topology Appl.
**52**(1993), no. 2, 127–149.

- Jeff Weeks,
Convex hulls and isometries of cusped hyperbolic 3-manifolds.
Topology Appl.
- Nov 3. More on canonical cell decompositions. Did 1 to middle of 4.
- Nov 5. Finding the canonical decomposition.
- Nov 8. Canonical decompositions in 3D; closed hyperbolic manifolds.
- Nov 10. More on closed manifolds.
- Nov 12. Hyperbolic Dehn filling. Through example on page 4.
- Nov 15. More on Hyperbolic Dehn filling. Did 1-4.
- Nov 17. Volumes of hyperbolic 3-manifolds. Did everything.
- 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. - Nov 29. Certifying solutions to gluing equations.
- Dec 1. The HIKMOT method.
- Zgliczynski, Notes on Krawczyk's test.
- HIKMOT, Verified computations for hyperbolic 3-manifolds.

- Dec 3. Applications of verified
hyperbolic structures.
- Dunfield, Hoffman, Licata, Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling.
- Dunfield, Floer homology, group orderability, and taut foliations of hyperbolic 3-manifolds, Section 6.

- Dec 5. Proof by parameter space.
- Dec 7. Machine learning for fun and profit.
- Davies, Juhász, Lackenby, Tomasev, The signature and cusp geometry of hyperbolic knots.
- Nature article.