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 9th. 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.
Notes from each lecture will be posted here.