So, I’m stuck in the Eugene airport for a few hours (the good news is that they have free wireless), due to weather in San Francisco (why a person going from Oregon to Boston would go through San Francisco, I’m not sure, but perhaps United Airlines can tell you). I’m finally done (assuming I actually get home today), with a crazy month of travelling, which included 6 different talks in 6 different places (admittedly in only 4 different cities). Of course, this lead to me putting my energies into making Beamer slides instead of writing posts (or papers), so I thought I would point any interested parties to said slides, which are now posted on my website.

The most recent ones are from a colloquium at the University of Oregon, with some generalities on knot homology and categorification, and then my thoughts on how to categorify Reshetikhin-Turaev knot invariants (this is still conjectural at the moment, but I think we’re getting closer).

Before that, I gave a talk at the University of Bonn (like I said, it was a crazy month) on the conjectures I’ve been working on with Braden, Licata and Proudfoot about symplectic duality. This is somewhat similar content to my MIT symplectic seminar talk, but with a much more algebraic focus (I think it was also a better structured talk, since I learned from some of my mistakes, but maybe that won’t come through just looking at the slides).

Even earlier, I gave a short talk at Knots in Washington about my work with Geordie Williamson on colored HOMFLYPT homology. This is an interesting story, in that this knot invariant has given a purely algebraic definition by Mackaay, Stosic and Vaz, but they haven’t given a proof of invariance, or that you get the right decategorification. Geordie and I have come up with a geometric description that allows you to prove invariance and decategorification. The paper is coming soon, I promise.

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Wow Ben. If your papers are half as good as your slide shows, then I’m impressed.

Those symplectic duality slides are great. I’m finally starting to understand what you are up to.

Some questions:

Do I understand correctly that the example you start with is one example of symplectic duality, the one coming from hypertoric varieties? The other examples of symplectic duality (such as category and Hilbert schemes of surfaces) do not come from hyperplane arrangements, right?

The subspace arrangement on slide 22 is not related to the hyperplane arrangements you discussed earlier, right?

And some nitpicks:

Slide 10: When you define a singularly symplectic space, you probably only want to require that be nonvanishing for

minimalresolutions. If I have any resolution , I can always make a new resolution by blowing up a point of ; will vanish of the exceptional fiber of .Slide 13: “GIT quotient of a variety with for the character .”

Slide 15: “This is quantum deformation …”

Slide 18: In traditional fraktur manuscripts, the “s” that looks like an “f” is not used as the last character of a word (and often not even as the last character of a syllable). See “events”, “necessary”, “dissolve”, “seperate and “station” here or “gesungen” and “alles” here.

re slide 18: Apparently the more historically correct form is: HIC ABVNDANT LEONES (with a drawing of a lion).

The propositions on slides 16 and 17 (existence of universal symplectic deformation and universal quantization) are really amazing! Are they written yet?

My first reaction was that they couldn’t be true, because deformation theory is really complicated. There are smooth surfaces whose deformation theory is arbitrarily bad.

There are surfaces whose holomorphic deformation space is a perfectly nice smooth ball, but where the algebraic surfaces form an countable collection of hypersurfaces in that ball.

After thinking about it a little bit, I realized that being the desingularization of a symplectic cone is pretty restrictive. For example, I tried to create examples by taking a smooth surface with complicated deformation theory, taking to be the cotangent bundle of and to be the contraction of the zero section in . But I found that the zero section was usually not contractible! The condition that the zero section be contractible in seems to be some sort of “extremely Fano” condition.

Anyway, that’s enough speculation from me. But I’d love to read a paper or a blog post about these issues.

David:

I should perhaps have been clearer in the slides (this is the sort of thing that is easy to say out loud, but forget to put in the slides): those theorems are not mine. The universal symplectic deformation is due to Kaledin and Verbitsky, and the quantization is a theorem of Bezrukavnikov and Kaledin.

The statement is a bit simpler for resolutions of cones, but the one for general holomorphic symplectic varieties isn’t substantially different. It only holds formally in general, and the fact that higher cohomology of the structure sheaf might not vanish makes trouble.

I’ll just note, these theorems are extremely unsurprising to those used to symplectic geometry: in the smooth context, the same theorems are true and not terribly difficult. The interesting thing is that one can make the complex structure move with you when you change the symplectic structure in such a way to keep it holomorphic.

David,

For an arbitrary projective variety X, the condition that T^*X is a resolution of Spec Fun(T^*X) is extremely restrictive–I believe that the only known examples are of the form G/P, and I’m not even sure that those work outside of Type A. This condition appears to be related to two other conditions, namely that T*X admits a complete hyperkahler metric, and that X is D-affine (some version of the statement that the category of D-modules on X is equivalent to the category of modules over the ring of differential operators).

In some sense, a large part of what Ben, Tom, Tony, and I are doing is trying to understand the relationship between these properties, as well as analogous ones in which T^*X is replaced by a more general symplectic variety.

Ben — thanks! Looks like I have a lot of reading to do.

David:

Your other criticisms are, of course, correct but with respect to the definition of “singularly symplectic”:

There’s a right criticism that could be made here and a wrong one, and I can’t quite tell which you are making, so let me just clarify in general.

The form doesn’t have to be nonvanishing, but it can only vanish on the exceptional set of the resolution. This condition doesn’t depend on which resolution you take; if such a form exists for one resolution, it exists for all.

On the other hand, it is true that I said “preimage of the singular set” when I should have said “exceptional set” and for that, I am deeply sorry.

Ah, I had the wrong one. I thought you were aiming for an analogue of crepant: a resolution of singularities is crepant if there is a volume form on the nonsingular part of whose preimage extends to a nonvanishing volume form on . (Well, that’s the definition when is a cone; in general, you want to say that has an open cover so that has the above property.)

I think I have another typo, though. In the paragraph defining a symplectic resolution, you write , when you mean the reverse.

Right, that’s a symplectic resolution.

My aim was: singularly symplectic = property of a singular space

symplectic resolution = property of a resolution (that’s the correct analogue of crepant, and in fact, a strengthening of crepant).

The only other plausible routing from Eugene to Boston on United would be through Denver, since I don’t imagine United has direct flights from Eugene to LAX, Chicago, or Washington, (just based on distance calculations) and I think those five exhaust their hubs. From there it just depends on what time and availability there is on the flights from Eugene to SFO or DEN, compared to flights from those places to Boston.

Believe me, Kenny, I’m well aware of this fact (and I did end going through Denver, because the only flight that could have gotten out of SF would have been the red-eye). It doesn’t mean I’m happy about it.

Over at the n-cat cafe, it was pointed out that printer friendly versions of slides are also desirable. Ben these slides are really interesting, could you reTeX them in

“\documentclass[10pt,handout]{beamer}”

that format?

Having made that request, I will also (eventually) provide printer friendly versions of talks for my own web page.

By the way, in case anyone else decides to be picky about Fraktur fonts, “s:” makes the “final s.”