Reshetikhin–Turaev Invariants

**Instructor:** Theo Johnson-Freyd

**Date and time:** MWF 3:00-4:00 in Lunt 101 (for now).

Occasionally I will send seminar announcements over email. To get on the list, please email me at theojf@math.northwestern.edu.

This seminar
will be on
**Reshetikhin–Turaev Invariants**
and their myriad relations to geometry, low dimensional topology, and
representation theory. Reshetikhin–Turaev invariants of surfaces and
three-manifolds were originally introduced in terms of Kirby moves and
surgery presentations, and were immediately recognized as a mathematical
construction of Witten's "quantum Chern--Simons theory." They also
arise in other ways, notably through skein theory and through geometric
quantization. A main ingredient, quantum groups, appeared originally in
the context of two-dimensional statistical mechanics. Much about
Reshetikhin–Turaev invariants is predicted by physics but mathematically
hard to establish: for example, the equivalence of the different
descriptions (which are automatically equivalent "at a physical level of
rigor") was only recently established; one of the deeper questions in the
field is a physical prediction relating certain asymptotics of these
invariants to the hyperbolic geometry of the corresponding three-manifold.
Reshetikhin–Turaev invariants also provide important tools for example
in the study of mapping class groups. Thus my goal in the course is to
use Reshetikhin–Turaev invariants as a window into many important areas
of research. Although I know a few of these stories well, I am by no
means an expert, and look forward to learning them. Conversely, I assume
that most of the audience are also novices.

The bulk of the course will consist of student lectures. I'll lecture the first four sessions, which leaves, if I counted correctly, 25 days. So each student participant will give about three lectures (*n*th years working on their theses may opt to lecture less than this).

For each lecture you give, please prepare typed lecture notes. They don't have to be flawless, but it's a good place to include full citations, etc. Your lecture notes will be posted on this website. At the end of the quarter, I'm hoping we'll edit together the different lecture notes into some sort of document expositing the material covered in the course.

If you are "young," then I expect your lectures will cover some classic paper on the subject. If you are "old," then I expect you to explain some part of your thesis work, setting your story within the context of the class.

Any student requesting accommodations related to a disability or other condition is required to register with AccessibleNU (847-467-5530) and provide professors with an accommodation notification from AccessibleNU, preferably within the first two weeks of class. All information will remain confidential.

Suspected violations of academic integrity will be reported to the Dean's Office. For more information on Northwesternâ€™s academic integrity policies, see http://www.weinberg.northwestern.edu/handbook/integrity/index.html.

**Monday, January 4:** (Theo) Statistical Mechanics and the Jones Polynomial. Notes.

**Wednesday, January 6:** (Theo) Statistical Mechanics and the Jones Polynomial. Notes.

**Friday, January 8:** (Theo) Statistical Mechanics and the Jones Polynomial. Notes.

**Monday, January 11:** (Theo) Statistical Mechanics and the Jones Polynomial. Notes.

**Wednesday, January 13:** (Theo) Statistical Mechanics and the Jones Polynomial. Notes.

**Friday, January 15:** (Theo) Statistical Mechanics and the Jones Polynomial. Notes.

**Monday, January 18:** *MLK Day*

**Wednesday, January 20:** (Theo) Statistical Mechanics and the Jones Polynomial. Notes.

**Friday, January 22:** (Piotr) Turaev--Reshetikhin surgery invariants of 3-manifolds

**Monday, January 25:** (Piotr) Turaev--Reshetikhin surgery invariants of 3-manifolds

**Wednesday, January 27:** (Piotr) Turaev--Reshetikhin surgery invariants of 3-manifolds

**Friday, January 29:** (Sean) Topological quantum field theory derived from the Kauffman bracket. Notes.

**Monday, February 1:** (Sean) Topological quantum field theory derived from the Kauffman bracket. Notes.

**Wednesday, February 3:** (Sean) Topological quantum field theory derived from the Kauffman bracket. Notes.

**Friday, February 5:** (Theo) Quantum SU(2) at finite level. Notes.

**Monday, February 8:** (Nilay) Some background on geometric quantization. Notes.

**Wednesday, February 10:** (Nilay) Some background on geometric quantization. Notes.

**Friday, February 12:** (Pyongwon) Geometric quantization of Chern-Simons gauge theory via Kähler polarizations. Notes.

**Monday, February 15:** (Pyongwon) Geometric quantization of Chern-Simons gauge theory via Kähler polarizations. Notes.

**Wednesday, February 17:** (Theo) Geometric quantization of Chern-Simons gauge theory via real polarizations. Notes.

**Friday, February 19:** (Theo) Geometric quantization of Chern-Simons gauge theory via real polarizations. Notes.

**Monday, February 22:** *Seminar canceled.*

**Wednesday, February 24:** (Theo) Braiding together the different constructions. Notes.

**Friday, February 26:** (Chris) Holomorphic Chern–Simons theory and the (super) twistor correspondence. Notes.

**Monday, February 29:** (Bif) Asymptotics, Groups, and Measures. Notes.

**Wednesday, March 2:** (Paul) Cobordism invariants and TMF. Notes.

**Friday, March 4:** (Ben) Higher enveloping algebras

**Monday, March 7:** (Aron) Conformal blocks. Notes.

**Wednesday, March 9:** (Eric) Genus-zero TQFTs and operads. Notes.

**Friday, March 11:** *Seminar canceled.*