In February there is going to be a workshop and school dedicated to exploring the interactions of Quantum Gravity, Higher Gauge Theory, and Topological Field Theory. I’m excited about the chance to share ideas and hopefully create some new mathematics.
The conference will take place in Lisbon, Portugal, and yours truly will be giving one of the mini-courses for the school (the topic is going to be the classification of extended 2D tqfts, something near and dear to my heart). Of course that makes me really excited, but I am also excited about the other topics too and I think the mix of ideas will be invigorating. For more info look below the break.
I wanted to take a moment to plug a conference in my soon-to-be hometown Eugene, OR organized by my once and future colleague Nick Proudfoot.
Aside from Eugene being lovely in August, I felt this conference was worth a post because it’s something of a unique format. Rather than being a bunch of experts on the subject (as it says in the title, the subject is the conjunction of operator algebras and CFT) getting together and giving talks that only they understand, it will be aimed at being educational for graduate students and interested non-experts (such as myself). The format is a bit similar to that of Talbot. In particular, in addition to an organizer (Nick) it has a “leader” who is in charge of mathematical content (but will delegate quite a few of the lectures); that will be the incomparable Andre Henriques. Continue reading
So I’ve finally managed to bang my dissertation into something more or less ready for public consumption. It is basically finished (except for some typos and spell checking).
It is available on my new website.
The title is “The Classification of Two-Dimensional Extended Topological Field Theories”.
This morning Jacob Lurie posted a draft of an expository paper on his work (with Mike Hopkins) classifying extended (infinity, n)-categorical topological field theories and their relation to the Baez-Dolan cobordism hypothesis.
Should make for some intersting bedtime reading…
This is the third and final post in my series about using planar algebras to construct TQFTs. In the first post we looked at the 2D case and came up with a master strategy for constructing TQFTs. In the last post we began carrying out that strategy in the 3-dimensional setting, but ran into some difficulties. In this post we will overcome those difficulties and build a TQFT.
In my last post I explained a strategy for using n-dimensional algebraic objects to construct (n+1)-dimensional TQFTs, and I went through the n=1 case: Showing how a semi-simple symmetric Frobenius algebra gives rise to a 2-dimensional TQFT. But then I had to disappear and go give my talk. I didn’t make it to the punchline, which is how planar algebras can give rise to 3D TQFTs!
In this post I will start explaining the 3D part of the talk. I won’t be able to finish before I run out of steam; that will have to wait for another post. But I will promise to use lots of pretty pictures!
So today I am giving a talk in the Subfactor seminar here at Berkeley, and I thought it might by nice to write my pre-talk notes here on the blog, rather then on pieces of paper destined for the recycling bin.
This talk is about how you can use Planar algebras planar techniques to construct 3D topological quantum field theories (TQFTs) and is supposed to be introductory. We’ve discussed planar algebras on this blog here and here.
So the first order of buisness: What is a TQFT?
I’ve been too lazy to write in detail about the progress in my research (well, I am writing six papers and applying to jobs, so it isn’t entirely due to laziness), but I did recently speak in the symplectic seminar at MIT, and have posted the slides on my webpage. Obviously, they’re less useful without someone to explain them, but given the current lack of an overarching paper on the subject (that’s no. 5 on the list, I promise), I thought it might be edifying. Executive summary below the cut. Continue reading
So, a subject rather near and dear to the hearts of many of my fellow co-bloggers is that of 1+1-dimensional TQFT: that is, of monoidal functors from the category of 1-manifolds with morphisms given by smooth cobordisms to the category of vector spaces over your favorite field .
There’s a rather remarkable theorem about such functors, which really deserves a post of its own for proper explanation, but I’ll spoil the surprise here.
Any such functor associates a vector space to a single circle, and to the “pair of pants” cobordism, it assigns a map , which one can check is a commutative multiplication.
Furthermore, the cap, thought of as a cobordism from the empty set to a circle gives a map , which gives a unit of this algebra. Thought of as a cobordism from the circle to the empty set, it gives us a map which we call the counit or Frobenius trace.
Theorem. A commutative algebra with counit arises from a TQFT if and only if kills no left ideal of .
Edward Witten gave two talks at MIT last week. The first was on gauge theory and wild ramification – very similar to earlier work he did with Kapustin and Gukov on geometric Langlands, but with some clever use of nineteenth century technology (namely, Stokes matrices) to deal with irregular singularities. I won’t say much about it, except to mention that his use of the term “wild ramification” employs a tacit conjectural dictionary between irregularity for differential equations and the Swan conductor for Galois representations. The second talk was on some calculations in pure 3D gravity he did with Alex Maloney, and even though I didn’t understand much of it, I’m going to write about it. Perhaps people with more background in phyisics or three-manifold topology can make illuminating comments and corrections.