Mathematics

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In April 2006, with Kevin Walker, I worked out how to fix the functoriality of Khovanov homology. What fun! I'm now writing up my thesis, and moving to Santa Barbara to join Microsoft's topological quantum computing group in April.

While in Hungary, April 2004, I began to understand how the construct the `Gelfand-Tsetlin' functor for sl_n spiders. I've been working out the consequences since then.

At the beginning of 2004 I was working on some things to do with A_n and D_n spiders, as well as thinking about foams and Khovanov categorification, and working on my never-ending Mathematica quantum groups project.

In 2003 I worked on a new proof of Penrose's `Spin Geometry Theorem', using symplectic geometry, and some ideas of Justin Roberts. It's on the backburner at the moment, but I'm keen to finish it off sometime. I also wrote a paper about calculating the coefficients appearing in the Jones-Wenzl projections. It hasn't been published, as I discovered that essentially similar results had previously appeared in a paper by Khovanov and Frenkel.

In May 2003 I passed my qualifying exam -- here's my syllabus and a list of qual questions.

Papers.

Skein theory for the D_2n planar algebras
Joint with Emily Peters and Noah Snyder, available at arXiv:0808.0764.
Fixing the functoriality of Khovanov homology
Joint with Kevin Walker, available at arXiv:math.GT/0701339.
On Khovanov's cobordism theory for su_3 knot homology
Joint with Ari Nieh, available at arXiv:math.GT/0612754. Accepted by Journal of Knot Theory and its Ramifications, May 24, 2007.
The Karoubi Envelope and Lee's Degeneration of Khovanov Homology
Joint with Dror Bar-Natan, Algebraic & Geometric Topology 6 (2006) 1459–1469. Available at DOI:10.2140/agt.2006.6.1459 or arXiv:math.GT/0606542.
Classifying Spinor Structures
BSc. honours thesis at UNSW. Available at arXiv:math-ph/0106007.

Talks.

2008

July 8, Introduction to Khovanov homology and genus bounds for links, Station Q seminar.

May 15, Blob Homology, Georgia Topology Conference.

April 18-20, Skein theory for the D_2n subfactors, Planar Algebras and Subfactors meeting at Vanderbilt.

February 1, Link invariants from the D4 subfactor, with Emily Peters and Noah Snyder, UC Berkeley [view=event&id=2605 subfactor seminar].

2007

December 12-15, Lasagna composition for Khovanov Homology, in the Quantum Topology Special Session of the joint NZMS/AMS Wellington meeting. (slides)

October 2, Spiders for U_q(sl_n), Station Q Seminar, UC Santa Barbara. (slides)

August 6-10, Lasagna composition for Khovanov Homology, Quantum Topology in Hanoi, Vietnam (slides)

July 5-8, Lasagna composition for Khovanov Homology, Oporto conference on Link Homology, Portugal (slides)

May 23, Functoriality and duality in Khovanov homology, Kyoto. (Nakajima's notes, photos)

May 15, Genus bounds and spectral sequences made easy, Kyoto. (slides)

May 14, Khovanov homology, Kyoto. (slide)

April 24, Khovanov homology is functorial in S^3, Braids In Banff. (slides)

February 21 and 22, Rasmussen's genus bounds from Khovanov homology, UC Berkeley and UC Davis topology seminars.

2006

December 4, A topological categorification of su_3, Colloquium, University of Oregon, Eugene. (slides)

December 1, Khovanov Homology, Basic Notions Seminar, University of Oregon, Eugene. (slides)

November 17, Disoriented and confused: fixing the functoriality of Khovanov homology. Columbia, Gauge Theory Seminar. (slides, handout.)

September 7, Disoriented and confused: fixing the functoriality of Khovanov homology. Uppsala, Categorification in Algebra and Topology. (slides, handout.) My pretty slides got a mention at the n-Category Cafe.

July 3, Disoriented and confused: fixing the functoriality of Khovanov homology. PCMI. (slides)

June 16, A cobordism theory for sl_3 knot homology, Subfactor seminar.

May 6, Disoriented and confused: fixing the functoriality of Khovanov homology. Knots in Washington. (slides, handout, actual handout)

April 28, Disoriented and confused: fixing the functoriality of Khovanov homology. (announcement, slides)

April 11, A cobordism theory for sl_3 knot homology (slides, handout), UCSB topology seminar.

Khovanov Homology, Taipa conference Jan 2006. (slides)

2005

omath, subfactor seminar.

The Knot Atlas, at Knots in Washington.

Bar-Natan's local Khovanov homology.

Generators and relations for $Rep(U_q(\mathfrak{sl}_n))$ as a pivotal category.

  • Abstracts February 11, February 18
  • I gave a short 25 minute version of this talk at the Quantum Topology AMS meeting at Snowbird, in June. You can download overheads as an ~8mb pdf.

2004

Pivotal categories, planar algebras and subfactors

2002

October 18 
The Temperley-Lieb algebra and $\mathfrak{sl}_2$ representation theory
October 11 
Jones-Wenzl projections in the Temperley-Lieb algebra.
May 8 
Juggling for Geeks (abstract) (in Many Cheerful Facts).

2001

June 8 
My honours talk at UNSW, Classifying Spinor Structures (abstract)

Reviews

  • A MathSciNet review of Plamenevskaya's paper, "Transverse knots and Khovanov homology".

UNSW.

I'm an alumnus of the University of New South Wales, in Australia. I graduated mid 2001 with a Bachelor of Science. I received the University Medal, and first class honours in the Pure Mathematics Department. Some articles I wrote as an undergrad at UNSW appear below. If you're interested, you can get copies here.

  • dvi pdf Connections on Principal Fibre Bundles.
  • dvi pdf Homotopy Invariance of Pullback Bundles (math.DG/0105161 at the mathematics e-print arXiv)
    • A preliminary version, with serious, but correctable, mathematical errors.
  • dvi pdf The Hodge Star, Poincare Duality and Electromagnetism.
  • dvi pdf Classifying Spinor Structures (math-ph/0106007)
    • My honours essay for the degree of Bachelor of Science at the University of New South Wales, Australia

Bibliography.

There's also my annotated bibliography [1], generated from the BIBTeX database of everything I read. Unfortunately I've stopped updating it, so it's quite out of date. There is also my bibliography page, containing brief bibliographies for various topics.