See below for a course description.
| Points at least (%) | Guaranteed grade |
|---|---|
| 97 | A+ |
| 93 | A |
| 90 | A- |
| 87 | B+ |
| 83 | B |
| 80 | B- |
| 77 | C+ |
| 73 | C |
| 70 | C- |
| 67 | D+ |
| 63 | D |
| 60 | D- |
This course is an introduction to the concept of quantum invariants and, more broadly, topological quantum field theories, which have been developed over the past three decades into a subject called quantum topology. The starting point for the subject is the discovery of the Jones polynomial of a knot and the formulation of it as a quantum field theory in physics in the 1980s. We will focus on the mathematical side of the theory and hence no physics background is required. Quantum invariants usually refer to invariants of knots/manifolds that are constructed in a similar style as the Jones polynomial, and are named so to contrast with classical invariants such as homology/homotopy groups.
A central problem in this area is the so-called volume conjecture that relates the colored Jones polynomial of a hyperbolic knot to the hyperbolic volume of the knot. There are a few versions of volume conjectures using different quantum invariants. Special cases of the conjectures have been verified but a full solution remains open.
In the first half of the course, we will define the colored Jones polynomial and generalize it to the Reshetikhin-Turaev invariants of 3-manifolds. Along the way, we will touch some topics on knot theory, skein theory, and tensor categories. In the second half, we will review some basics in hyperbolic geometry in order to formulate the volume conjecture. Towards the end, we will sketch the proof for some special cases of the conjecture. The course is aimed to be self-contained and some basic backgrounds in knot/3-manifold theory, category theory, and differential geometry will be useful, but not strictly required.
Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are welcome to let me know so that we can discuss options. You are also encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247.
If you have been certified by the Disability Resource Center (DRC) as eligible for accommodations, you should contact your instructor to discuss your accommodations as soon as possible. Here are instructions for sending your Course Accessibility Letter to your instructor: https://www.purdue.edu/drc/students/course-accessibility-letter.php