For those of you still wondering whether I’ve made it home yet, the answer is: nope. I’m in Freiburg, Germany, to try and get a little work done with my friend and collaborator Geordie Williamson (keep an eye peeled for our paper on the arXiv, though it’s actually been posted on my webpage for a month now). We seem to have made some good progress, both on this mostly done paper and some more speculative projects (I can’t reveal any details, but there were affine Grassmannians involved), and I’m giving a talk tomorrow to edumacate the algebra-types around here about knot theory a little (it’s one of those weird perspective twists that comes from being at Berkeley that I tend to expect algebraists to know some rough version of the Jones polynomial story and some knot theory basics, which is clearly insane).
The rough sketch of my talk is as follows:
- What is a knot? What are knot invariants? What good are they?
- The construction of the Jones polynomial and HOMFLY polynomial, and their connection to representation theory.
- How Khovanov-Rozansky homology using Soergel bimodules categorifies this story, step by step.
- The geometric version of this story, including the results from my paper with Geordie (in about 3 seconds at the end).
We’ll see how this goes. On one hand, I apparently can take up to an hour and 45 minutes (this seems to be standard in Freiburg). On the other, Sebastian Zwicknagl will be giving a hour and a half or so talk starting at 11, we’ll have lunch and then launch directly into my talk, so I may eventually take pity on my audience.
It’ll be a little strange, in that usually one of the hardest parts of this sort of talk is convincing the audience that Soergel bimodules are not terrifying and wierd (made more difficult by the fact that it is patently false). Whereas this time Soergel will be there and close to half the audience will be his mathematical descendants. This will make the algebra easier, but presumably this will lead to the knot theory being a bit more of a struggle than usual.