TQFTs via Planar Algebras (Part 2)

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!

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TQFTs via Planar Algebras

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?

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Generalized Homology Theories

Recently there have been some comments on our requests page:

Sander Kupers:

Maybe you could explain a bit about elliptic cohomology and topological modular forms…

and Thomas Riepe:

I would be curious about learning more on:
“… many constructions of classical algebra (eg, the theory of modular forms) are beginning to be seen to have deep homotopy-theoretic foundations.”…

Since this is somewhat related to some of my research, I have been recruited volunteered to talk about these things. The problem is that this is a Huge subject and a difficult subject and there is no way to adequately represent it in blog form. On the other hand it is a beautiful subject, filled with lots of exciting tidbits. Why not give it a go anyway, right?

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A Bicategory of Groupoids

I want to talk about an interesting 2-category of topological groupoids that I’ve been thinking about recently. Let’s start with the basics; what is a groupoid? Well a group can be thought of as a category with one object and with every morphism invertible. A groupoid is the same thing, except that there can be multiple objects. Why are these interesting? well groupoids generalize three different notions at once: sets, equivalence relations, and groups. We’ve seen how groups enter. Equivalence relations on sets can also be viewed as categories where there are exactly one or zero morphisms between two objects: one if the objects are equivalent, zero otherwise.

Obvious Fact: Since groupoids are categories we can talk about functors between groupoids and natural transformations between these. Hence groupoids form a 2-category.

What happens if we add topology? What if we want a space of objects and a space of morphisms? (source, target and identiy maps are countinuous, of course). Well things start to break down.

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Spheres in Higher Dimensions

I just reread this problem on the Harvey Mudd Math Fun Facts website. It’s about your basic intuition in higher dimensions. I remember when I first heard it I was so shocked.

Suppose you have a square in the plane. Cut it into four quadrants, then inscribe a circle in each. Finally add a fifth circle which is just tangent to the other circles.

Square with inscribed circles.

As you can see, it is much smaller then the other circles and the square.

You can repeat this in higher dimensions. For example, you can cut the cube into octants and inscribe eight spheres in these, then add a last final sphere, tangent to the eight inscribed spheres.

So how does the volume of this final sphere change with respect to the volume of the cube when we change dimensions? How does its size compare? It’s always smaller right? WRONG!

As you increase dimension, the final sphere gets bigger and bigger. In dimension 9 it is tangent to the original n-cube and in dimensions bigger than 9 it pushes outside the original cube. Eventually as the dimension increases, the volume of the final sphere exceeds that of the n-cube!

The details are pretty easy and can be found here:
Su, Francis E., et al. “High-Dimensional Spheres in Cubes.” Mudd Math Fun Facts. http://www.math.hmc.edu/funfacts.

There are all sorts of other cool math fun facts on this webpage. You should check it out.