Spring 2021 MATH 595
Quantum, Complexity, and Topology (QCaT)


Low-dimensional topology has seen significant advances since the 1980s, largely along two parallel tracks. On one hand, in the early 2000s Perelman proved Thurston's geometrization conjecture, a vast generalization of the Poincaré conjecture that serves as a kind of theoretical classification of 3-dimensional manifolds. On the other hand, topology and physics have cross-pollinated significantly via ideas like field theory, supersymmetry, and quantum mechanics. In some ways, the mathematical theory undergirding each track has been so thoroughly developed that further advances will probably be best understood using the language of theoretical computer science and computational complexity.

For instance, topologists now know that there exists an algorithm to solve the homeomorphism problem for 3-manifolds, but very little is known about the precise computational complexity of it. Meanwhile, physicists motivated by quantum computing, quantum error-correction, or condensed matter theory sometimes want to understand the complexity of approximating certain invariants of 3-manifolds. At the center of all this is a beguiling and unanswered question: are quantum computers any better at topology than classical computers?

The goal of this class is to systematically develop the connections between topology and computer science, especially quantum computing. Our perspective will be motivated primarily by topology, although we will aim to make the material accessible and useful to those with physics or CS backgrounds too. In the first part of the class, we will introduce the basics of geometric topology, theoretical computer science, and quantum physics and quantum algorithms. We will then turn to some deeper topics: the complexity of unknot recognition, error-correcting codes, topological quantum computing and the hardness of quantum 3-manifold invariants, and, time permitting, the role played by finite groups in TQFT invariants, and categorified invariants of knots such as Khovanov homology.


There are no required course textbooks, but you will find some links to relevant papers on the course calendar. My lectures notes from class will also be uploaded there.

If you want to build more background on the things we discuss in class, the following are some helpful resources and, when I could find them, links to their electronic access through the Illinois library. (Note that a university log-in is required. I recommend you download the pdfs, as I have already done for myself. If you need help getting access to any of them, let me know.)


Enrolled students will be graded on participation and engagement. They will be expected to attend (virtual) lectures regularly or, with my permission, watch the recordings instead. Students may exhibit their engagement by giving a talk on a paper or theorem (I will announce these opportunities as we go), writing an expository article, submitting some solutions to exercises I indicate in class, or other options. I am open to suggestions.

Communication plan

I don't expect for us to have to communicate much outside of class, but, when we do, let's use email. I will try to reply within 24 hours on weekdays, but will often not reply over weekends.


All of the regularly scheduled course meetings will be recorded and made publicly available on the course Media Space channel, and possibly other places too (e.g. YouTube). By default, cameras and microphones will be turned off when you join the Zoom meeting, but you are strongly encouraged at least to turn your mic on when you have questions or want to discuss. Student presenters will be able to decide for themselves whether they want to be recorded or not.

I'd like to have some control over how many people join our meetings, so please do not share the Zoom link and password with anybody not enrolled in the class without my explicit permission.

Reasonable accommodations

Any students with disabilities who require reasonable accommodations should please contact both me and DRES as soon as possible. There won't be any exams or anything like that, but, e.g., if the videos, course notes, or above textbooks are not accessible for some reason, I would love to be able to fix that.