# New Photograph

Last Friday, we had a seminar at Berkeley — or rather, at Noah’s house — featuring Mike Freedman and some quantity of beer. Mike spoke about some of the hurdles he had to overcome in writing his recent paper with Danny Calegari and Kevin Walker. One of the main results of this paper is that there is a “complexity function” c, which maps from the set of closed 3-manifolds to an ordered set, and that this function satisfies the “topological” Cauchy-Schwarz inequality.

$c(A \cup_S B) \leq max \{c(A \cup_S A),c(B \cup_S B)\}$

Here, $A$ and $B$ are 3-manifolds with boundary $S$. [EDIT: and equality is only achieved if $A = B$] This inequality looks like the sort of things you might derive from topological field theory, using the fact that $Z(A \cup_S B) = \langle Z(A), Z(B) \rangle_{Z(S)}$. Unfortunately, it’s difficult to actually derive this sort of theorem from any well-understood TQFT, thanks to an old theorem of Vafa’s, which states roughly, that there’s always two 3-manifolds related by a Dehn twist that a given rational TQFT can’t distinguish. Mike speculated that non-rational TQFT might be able to do the trick, but what he and his collaborators actually did was an end run around the TQFT problem. They simply proved that that the function $c$ exists.

I tell you all this, not because I’m about to explain what $c$ is, but to explain our new banner picture. We realized after the talk that there were a fair number of us Secret Blogging Seminarians in one place, and that we ought to take a photo.

# Freedman on distinguishing manifolds with quantum topology

This week Mike Freedman was in Berkeley for the annual series of three Bowen lectures. The first two were about topological quantum computation and the fractional quantum Hall effect. Since I missed one of those because of closing arguments in the case I was on the jury for, I’ll instead only discuss his third talk which dove-tails nicely with my last post.

His motivating question was why do we look for topological quantum computation in 3 space-time dimensions (that is, by restricting to a system stuck in a plane) rather than the full 4 space-time dimensions? His explanation was the following: In 3-dimensions it seems that unitary TQFT can distinguish all manifolds, however in 4-dimensions they can not.