Well, contrary to my (and everyone else’s) expectations, I did not, in fact, give my talk on the geometry of Soergel bimodules today. That’s right, I slept in. I was actually only 10 minutes late for the talk, but that was late enough that they had started another speaker in the interests of continuing flow. On the whole this is not such a bad result. Slightly mortifying for me, but a passing blip for most attendees. The only really annoying part is that I was rescheduled to give the last talk of the conference (this was the slot of Dylan Thurston, who ended up talking), which everyone knows is the worst slot imaginable. Everyone is exhausted by that point, at least the ones who haven’t already left, and if anyone has comments on your talk, you have no time to discuss them, since they have a plane to catch. I recognize that this is entirely my own fault, but really, doesn’t that just make it that much more annoying?
Anyways, here’s a little about the talks I did see:
Dylan Thurston gave a pretty good talk about combinatorial knot Floer homology (a topic many knot theorists are rather excited about). It was a lot of stuff I had already seen, but did have some interesting new stuff about using knot Floer homology, and some canonical elements in it, to distinguish transverse knots with the same underlying knot structure. Not the sort of knot theory I tend to get excited about, but it does seem like a rather cool technique.
Adam Sikora gave a talk about some very interesting stuff relating quantum invariants of knots to more classical properties of knots. It’s not stuff that I’m too expert in, but here’s my quick summary: If you have any knot K, then one can take a torus that fits around the knot like a tube. This defines a homomorphism . Let be the character variety of for some algebraic group G (that is the set of closed G-orbits on the set of homomorphisms from to G, with the conjugation action). In good cases (I believe G reductive is good enough), the set of such homomorphisms factoring through will define a Lagrangian (in particular, half-dimensional) subvariety of .
Now, there are a couple of natural ways of constructing a deformation quantization of . One of these involves an algebra of operators on the space of -valued functions on the weight lattice of $G$ (essentially, the Weyl group invariants inside a quantized Weyl algebra), and another involves skein modules of surfaces (this only works in . Sikora conjectures that certain natural ideals in these quantizations (in the former, the ideal of operators killing the quantum invariants of a fixed knot K, thought of as functions on the weight lattice, in the latter a similar ideal, defined via a geometric action using the knot K) will reduce to the ideal of functions vanishing on . Various versions of these have been proven for , but the higher rank picture still has a lot of mysteries.
Understanding the connections between quantum and classical knot invariants is one of the great challenges facing knot theory in the 21st century, and some really exciting advances are being made. Maybe it’s my own prejudice toward deformation quantizations of character varieties, but this strikes me as a really beautiful one.
Anna Beliakova gave a talk on a simplification of combinatorial Heegard-Floer homology (or perhaps it should be called a complication that eases computation) that sounded rather exciting (I’m all in favor of the size of complexes dropping by orders of magnitude), but I don’t have the expertise to comment, past a vague enthusiasm for the more efficient computation of knot invariants, especially by identifying redundant summands in chain complexes.