Spring 2017 Learning Seminar

This is the webpage for a learning seminar on quantum physics and categorification in low-dimensional topology. In Spring 2017, we worked through two extended examples: Witten's derivation of the Jones polynomial from Chern-Simons theory and Khovanov's categorification of the Jones polynomial into Khovanov homology. The seminar was organized by Arun Debray and Richard Wong.

**Logistics:** we met **Tuesdays** at **2:00 PM** in room **RLM 11.176**. We tackled the Jones
polynomial and Chern-Simons theory in the first half of the semester, then moved to Khovanov homology.

**Lecture notes** for this seminar are
located here.

**Schedule:**

January 18, 2017: organizational meeting

January 24, 2017 (Hannah): Knot polynomials and skein relations

January 31, 2017 (Ivan): Introduction to quantum field theory, part 1 [PS95], [QFS2] Lecture 1

February 7, 2017 (Jay): Introduction to quantum field theory, part 2 [PS95], [QFS2] Lecture 1

February 14, 2017 (Arun): Witten's derivation of the Jones polynomial, part 1 [Wit89]

February 21, 2017 (Sebastian): Witten's derivation of the Jones polynomial, part 2 [Wit89]

February 28, 2017 (Gill): Chern-Simons through surgery theory, part 1 [BHMV93]

March 7, 2017 (Adrian): Chern-Simons through surgery theory, part 2 [BHMV93]

March 14, 2017: spring break, no talk

March 21, 2017 (Jacob): Construction of Khovanov homology; relationship to the Jones Polynomial [DBN02], [Tur06] Lecture 1

March 28, 2017 (Allison): Applications of Khovanov homology [Tur06] Lecture 4

April 4, 2017 (Yixian): Khovanov homology for tangles; Frobenius algebras; cobordism categories [DBN05], [Tur06] Lecture 2

April 11, 2017 (Richard): Relationship to cobordism categories and TQFT [DBN05], [Tur06] Lecture 2

April 18, 2017 (Tom): LES, Functoriality, Invariants [Tur06] Lecture 3

April 25, 2017 (Christine): More applications: A transverse link invariant [Pla04]

**Main references:**

[BHMV93] C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological Quantum Field Theories Derived From The Kauffman bracket.

[DBN02] Dror Bar-Natan, On Khovanov’s categorification of the Jones polynomial.

[DBN05] Dror Bar-Natan, Khovanov's homology for tangles and cobordisms.

[Pla04] Olga Plamenevskaya, Transverse knots and Khovanov homology.

[PS95] Michael Peskin and Daniel Schroder,

*An Introduction to Quantum Field Theory*.[QFS2] Kryzysztof Gawędzki, “Lectures on Conformal Field Theory,” part 3, volume 2 of

*Quantum Fields and Strings: A Course for Mathematicians*.[Tur06] Paul Turner, Five Lectures on Khovanov Homology.

[Wit89] Edward Witten, Quantum field theory and the Jones polynomial.

**Related links:**

Paul Turner, A Hitchhiker's guide to Khovanov homology.

Scott Carter (Mathoverflow), What precisely is categorification?

John Baez and James Dolan, Categorification.

David Rose, A Note on the Grothendieck Group of an Additive Category.

Edward Witten, Khovanov Homology And Gauge Theory.

Scott Morrison and Ari Nieh, On Khovanov's cobordism theory for 𝔰𝔲(3) knot homology.

Kevin Walker, On Witten's 3-manifold Invariants.

Dan Freed, Remarks on Chern-Simons theory.

Nicolai Reshetikhin and Vladimir Turaev, Invariants of 3-manifolds via link polynomials and quantum groups.

Nicolai Reshetikhin, Lectures on quantization of gauge systems.

Papers on Khovanov homology and homotopy theory: Lipshitz-Sarkar, Everitt-Turner, Hu-Kriz-Kriz.

If you'd like to be on the email list, please send me an email.