Quantum Topology and Categorification

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 PMA 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.

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