# Conference on Higher Gauge Theory, Quantum Gravity, and Topological Field Theory

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.

# 20 questions

Two Berkeley grad students, Pablo Solis and Andrew Critch, just organised a “20 questions seminar”. The premise is pretty simple — everyone gets 2 minutes to ask a question they’d like an answer to, and we spend any remaining time (and tea afterwards) talking about them.

I wasn’t sure beforehand whether or not it was going to work out, but ended up pretty pleased I came. We didn’t quite make it to 20 questions (13 that I counted); they appear below. There’s also apparently going to be a wiki page for the seminar. The questions range from easy to weird to awesome. I left out one or two for various reasons, and my apologies for the lame TeX. Decide for yourself which you like, and feel free to give answers — if there’s good discussion here I’ll advertise that at next week’s “20 questions”.
1 Scott:
in R^2, you can tile the plane with hexagons. However any closed trivalent graph has a face that’s smaller than a hexagon. You can tile R^3 with vertex-truncated octahedrons. Say we have a “generic” closed finite cell-complex (every edge has 3 incident faces, every vertex has 4 incident edges). Is there something “smaller” than a vertex-truncated octahedron (or the other polytopes that give generic tilings)?

2 Critch:
Is there a space with trivial homology, non trivial homotopy?
(Anton: isn’t there a result that say that first nontrivial homology and homotopy agree?)

4 Yael:
Out(G) = Aut(G)/Inn(G). Is there a nice description of cosets, beyond that they’re cosets?

5 Mike:
X a banach space, f:X->R convex.
If X is infinite dimenionsal, what extra conditions guarantee that f is continuous?

6 Darsh:
Take a triangle in R^2 with coordinates at rational points. Can we find the smallest denominator point in the interior? (Take the lcm of the denominators of the coordinates.)
(Hint: you can do the 1-d version using continued fractions.)

7 Jakob
Take a “sparse” (every vertex has reasonably small degree) graph. Consider a maximal independent set for the graph (a maximal set of disconnected vertices). Can we make a new graph, with vertices the set, and whatever edges we like, that is as topologically similar to the original graph as possible?

8 Andrew:
What’s the deal with algebraic geometry? Just kidding!
Consider the sequence x0=0, x1=1, x_{n+2} = a x_{n+1} + b x_n, generalizing the Fibonacci sequence. Fix p a prime. If k is minimal so p|x_k and p|x_l implies k|l, then v_p(x_nk) = v_p(x_k) + v_p(n). (Here v_p(z) is the power of p dividing z.) Is there some framework that makes this sort of result obvious? Andrew only knows strange proofs.

9 Anton:
Take I=[0,1), the half open interval. Do there exist topological spaces X and Y, with X and Y not homeomorphic, but XxI and YxI are homeomorphic?
E.g., if instead I=[0,1], the closed interval, you can take X=mickey mouse=disc with two discs removed, Y=cross-eyed frog=disk with two linked bands glued on the boundary.

10 Pablo:
x^x^x^x … converges if x \in [e^{-1}, e^{1/e}], e.g. with x=\sqrt{2}, this converges to 2. Given a sequence (a_i), when does the “power tower sequence” converge?

13 Andrew again:
Can you define the set of all primes with a first-order theory?

Given a first-order theory, let S(T)={|M| | M is a finite model of T}.
You can get all prime powers with the field axioms. Is there some T so S(T)={primes}.

14 Yuhao:
Let A be an abelian category, that might not have enough injectives? Can you embed into another abelian category with enough injectives? Is there a universal way?
e.g. finite abelian groups embeds into Z-modules
e.g. coherent sheaves embeds into quasi-coherent sheaves
(Anton: the Freyd embedding theorem says every abelian category embeds in Z-mod. But this doesn’t help universality.)

# Mike on Topological Quantum Computing, at Georgia

I’m here at the 2009 Georgia Topology Conference and Mike Freedman is about to start talking about the current proposal for building a topological quantum computer. I’ll try liveblogging his talk; there’s a copy of the slides at http://stationq.ucsb.edu/docs/Georgia-20090518.pptx (PowerPoint only, sorry!) if you want to see the real thing. I think he recently gave a version of this talk in Berkeley recently, so some of you may have already heard it. I’ll fail miserably at explaining everything he talked about, but ask questions in the comments!

Mike says that the point of the talk will be to explain how it is that there’s a “topological” approach to building a computer, and try to give an idea of the mathematics, physics and engineering problems involved.

# Tikz, Slides, and Dissertations

There has been some recent discussion about tikz and beamer and I wanted to throw my two cents into the mix. What better way then by showing off the slides I used last week to talk about my dissertation? I had a lot of pictures, all of which I made entirely in tikz. Here is the link.

# More slides

My tendency to write slideshows instead of actual posts continues. If you like to see oodles of subtle variations on the same talk, you can see my slides from speaking at ARTIN in Glasgow (which just happened to be coincidentally scheduled during the breaks of the categorification conference there), which is the 8th time I’ve given that talk this year (I’m giving a talk today which will be my 13th total talk of 2009. You can see why I’ve been spending more time with Beamer than on the blog).

However, if you’re looking for something newer, this time you have a chance to see the slideshow before the people coming to the talk. I’m speaking on my work with Geordie in about 45 minutes, and made a Beamer show to accompany part of the talk.

Notably, this is the first Beamer I’ve made with Tikz. I’m particularly proud of the picture on slide 17, which I’ve posted under the cut:

# The Witt group, or the cohomology of the periodic table of n-categories

A very popular topic at the Modular Categories conference was the a generalization of the Witt group which is being developed by Davydov, Mueger, Nikshych, and Ostrik. What is this Witt group? Well it’s the simplest case of the cohomology of the periodic table of n-categories!

In this post I want to explain the definition of this cohomology theory and explain why it generalizes the classical Witt group.

First recall the Baez-Dolan periodic table.

Periodic Table

# Extended Haagerup Exists!

Following on from Noah’s post about the great Modular Categories conference last weekend in Bloomington, I’ll say a little about the talk I gave: Extended Haagerup exists!

The classification of low index, finite-depth subfactor planar algebras seems to be a difficult problem. Below index 4, there’s a wonderful ADE classification. The type A planar algebras are just Temperley-Lieb at various roots of unity (and so the same as $U_q(sl_2)$, as long as you change the pivotal structure). The type D planar algebras (with principal graphs the Dynkin diagrams $D_{2n}$) were the subject of Noah’s talk at the conference, and the $E_6$ and $E_8$ planar algebras are nicely described in Stephen Bigelow’s recent paper.

But what happens as we go above index 4? In 1994 Haagerup gave a partial classification up to index $3+\sqrt{3} \equiv 4.73205$. He showed that the only possible principal graphs come in two infinite families

and

(in both cases here the initial arm increases in steps of length 4) and another possibility

This result really opened a can of worms. Which of these graphs are actually realised? (Hint, they’re nicely colour-coded :-) What about higher index? What does it all mean? Are these graphs part of some quantum analogue of the classification of finite simple groups? Read one for the answer to the first question, at least.

# Coincidences of tensor categories

This week Scott and I were at a wonderful conference on Modular Categories at Indiana University. I find that I generally enjoy conferences on more specific subjects, especially in algebra. Otherwise you run the danger of every talk starting by defining some algebra you’ve never heard of (and won’t hear of again the rest of the conference) and then spend a while proving some properties of this random algebra that you still don’t know why you care about (let alone why you should learn about its projective modules). With more specific conferences if you don’t quite get something the first time you have a good change of seeing it again and it slowly sinking in. The organizers (Michael Larsen, Eric Rowell, and Zhenghan Wang) did an excellent job putting together and interesting, topically coherent, and fun conference. I was also pleasantly surprised by Bloomington, which turned out to actually be kind of cute. I have several posts I’d like to give on other people’s talks, in particular there were several talks (by Davydov, Mueger, and Ostrik) on the “Witt group” which involves the simplest case of a kind of cohomology of the periodic table of n-categories and thus should appeal to all of you over at the n-category theory cafe. But I think I’ll start out with our talks (which Scott and I prepared jointly based on our joint work with (Emily Peters)^2 and Stephen Bigelow).

The first of these talks (click for beamer slides) was on coincidences of small tensor categories. The strangest thing about this talk was that I was introduced as a “celebrity math blogger.”

Please note that in the slides I’ve completely elided the distinctions between a quantum group, its category of representations, and (when q is a root of unity) its semisimplified category of representations (where you quotient out by the kernel of the inner product as in David’s post).

# Slides-o-rama

So, I’m stuck in the Eugene airport for a few hours (the good news is that they have free wireless), due to weather in San Francisco (why a person going from Oregon to Boston would go through San Francisco, I’m not sure, but perhaps United Airlines can tell you).  I’m finally done (assuming I actually get home today), with a crazy month of travelling, which included 6 different talks in 6 different places (admittedly in only 4 different cities).  Of course, this lead to me putting my energies into making Beamer slides instead of writing posts (or papers), so I thought I would point any interested parties to said slides, which are now posted on my website.

The most recent ones are from a colloquium at the University of Oregon, with some generalities on knot homology and categorification, and then my thoughts on how to categorify Reshetikhin-Turaev knot invariants (this is still conjectural at the moment, but I think we’re getting closer).

Before that, I gave a talk at the University of Bonn (like I said, it was a crazy month) on the conjectures I’ve been working on with Braden, Licata and Proudfoot about symplectic duality.  This is somewhat similar content to my MIT symplectic seminar talk, but with a much more algebraic focus (I think it was also a better structured talk, since I learned from some of my mistakes, but maybe that won’t come through just looking at the slides).

Even earlier, I gave a short talk at Knots in Washington about my work with Geordie Williamson on colored HOMFLYPT homology.  This is an interesting story, in that this knot invariant has given a purely algebraic definition by Mackaay, Stosic and Vaz, but they haven’t given a proof of invariance, or that you get the right decategorification.  Geordie and I have come up with a geometric description that allows you to prove invariance and decategorification.  The paper is coming soon, I promise.

# TQFTs via Planar Algebras (Part 3)

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.