The meaning of knot homology October 1, 2009
Posted by Ben Webster in link homology, low-dimensional topology.8 comments
What I left out of my post on knot homology was, perhaps, the elevator pitch (if you’re in an elevator with a mathematician who already has some background). I’m giving a talk tomorrow which should include this stuff, and so one possibility is to just drop the slides for that on you, and let that speak for itself. Especially recommended are slides 12 and those past 18 (the rest is more standard quantum topology and categorification stuff).
But I’m not so sure that’s a wise plan. So let me try to say something more bloggy:
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A hunka hunka burnin’ knot homology September 24, 2009
Posted by Ben Webster in Category Theory, category O, combinatorics, homological algebra, link homology, low-dimensional topology, quantum groups, representation theory.19 comments
One of the conundra of mathematics in the age of the internet is when to start talking about your results. Do you wait until a convenient chance to talk at a conference? Wait until the paper is ready to be submitted to the arXiv (not to mention the question of when things are ready for the arXiv)? Until your paper is accepted? Or just until you’re confident you’ve disposed of any major errors in your proofs?
This line is particularly hard to walk when you think the result in question is very exciting. On one hand, obviously you are excited yourself, and want to tell people your exciting results (not to mention any worries you might have about being scooped); on the other, the embarrassment of making a mistake is roughly proportional to the attention that a result will grab.
At the moment, as you may have guessed, this is not just theoretical musing on my part. Rather, I’ve been working on-and-off for the last year, but most intensely over the last couple of months, on a paper which I think will be rather exciting (of course, I could be wrong). (more…)
Man and machine thinking about SPC4 June 29, 2009
Posted by Scott Morrison in crazy ideas, link homology, low-dimensional topology, papers.11 comments
I’ve just uploaded a paper to the arXiv, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, joint with Michael Freedman, Robert Gompf, and Kevin Walker.
The smooth 4-dimensional Poincaré conjecture (SPC4) is the “last man standing in geometric topology”: the last open problem immediately recognizable to a topologist from the 1950s. It says, of course:
A smooth four dimensional manifold 
We try to have it both ways in this paper, hoping to both prove and disprove the conjecture! Unsuprisingly we’re not particularly successful in either direction, but we think there are some interesting things to say regardless. When I say we “hope to prove the conjecture”, really I mean that we suggest a conjecture equivalent to SPC4, but perhaps friendlier looking to 3-manifold topologists. When I say we “hope to disprove the conjecture”, really I mean that we explain an potential computable obstruction, which might suffice to establish a counterexample. We also get to draw some amazingly complicated links:
Mike on Topological Quantum Computing, at Georgia May 18, 2009
Posted by Scott Morrison in conferences, low-dimensional topology, quantum computing, talks.12 comments
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.
Extended TFTs May 13, 2009
Posted by Chris Schommer-Pries in Paper Advertisement, QFT, Shamelss Self Promotion, differential geometry, low-dimensional topology, mathematical physics, tqft, websites.10 comments
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”.
How to get an algebra from a knot invariant April 13, 2009
Posted by Ben Webster in low-dimensional topology.4 comments
So, a couple of months ago, I gave a talk at the Max Planck Institute on knot homology, and as motivation I tried to explain why anyone studying the HOMFLY polynomial is inexorably led to the Hecke algebra. Nathan Geer, who was in the audience, asked me afterwards if there was anywhere this construction was written down, and lacking a good answer or the ambition to write a paper about it myself, I thought I would try to explain it in a blog post. It’s just applying an old TQFTologist’s trick, but old tricks often still have some new life in them.
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TQFTs via Planar Algebras (Part 3) December 10, 2008
Posted by Chris Schommer-Pries in Pictorial Algebra, QFT, low-dimensional topology, planar algebras, quantum algebra, subfactors, talks, tqft.add a comment
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.
TQFTs via Planar Algebras (Part 2) December 6, 2008
Posted by Chris Schommer-Pries in Pictorial Algebra, QFT, low-dimensional topology, planar algebras, subfactors, talks, tqft.5 comments
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!
TQFTs via Planar Algebras December 5, 2008
Posted by Chris Schommer-Pries in Pictorial Algebra, QFT, low-dimensional topology, planar algebras, subfactors, talks, tqft.7 comments
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?
More mathematical confections June 8, 2008
Posted by Ben Webster in differential geometry, low-dimensional topology, small examples.7 comments
The Poincaré disk has never looked so delicious. (more…)
