Its now been several days since the conference at IAS, and I might as well do a quick wrap up. Overall, the conference was great. Almost all the talks were understandable, and contained new mathematics of a truly humbling quality. There was a surprising collection of internet mathies: Ben Webster, Joel Kamnitzer, Peter Woit, Charles Siegel, Aaron Bergman and David BenZvi (and maybe even some I’m forgetting or didn’t notice). Also, I didn’t actually play Bad Talk Bingo; I had more than enough of my own math to do during downtime and bad talks. It was standing room only for virtually every talk (though that was at least in part because we were in a small room when a big auditorium was being unused). I even sat in the aisle next to Deligne for the first talk.
One downside is that it was a pretty intimidating atmosphere for asking questions. Usually, I’m pretty good about asking potentially stupid questions, but I wasn’t confident in my knowledge of the ‘basics’. I asked some of the speakers my questions after the talks, and I tried to write them all down for further contemplation. I figured I’d put ’em here, both for my own reference and in case anyone out there knows the answer. I’ve included the name of the talk the question is from; but in some cases, these are unrelated questions I was thinking about.

Vezzosi – One theme he emphasized was that, in derived algebraic geometric, the information carried in the higher homotopy groups is like a ‘nilpotent’ part of the scheme. My general question is: how far can one take this analogy? Specifically, given a (nonderived) scheme , can one construct the category of derived schemes whose is (similar to Artinian schemes over )? Can one define a derived version of ‘prorepresentability’ for this category? Presumably, this requires looking at the indcompletion of this category, which would be the derived analog of a formal scheme? Since moduli spaces of schemes are often more naturally derived schemes than schemes, I’m wondering if ‘derived prorepresentability’ is somehow nicer than prorepresentability. I asked Vezzosi about it after his talk, and he seemed to affirm most of these questions, saying that Jacob Lurie has addressed them.

Costello – He mentioned in passing that any time we have a complex/dga (whose differential is of degree +1), and we equip it with a squarezero derivation which commutes with and is of degree 1, that this amounts to giving the complex/dga a circle action. I understand why giving something a grading gives it a circle action, but not for a mixing differential.

Okounkov – He mentioned the quantum CalogeroMoser equation (I think). I know and care enough about the classical CM equation that I am curious as to what it looks like. Anybody know a resource?

Seidel – He took with marked points, and the branched double cover over those points. By connecting the marked points into a tree, and looking at the preimage, he got a sublocus whose Floer homology generated the Floer homology of the total space. The multiplication here corresponded to thinking of this doubled tree as a quiver and giving is some relations. This whole process was eerily reminiscent of ‘doubling a quiver’, even though the relations seemed a bit different. Is there any relation? (Warning: my comprehension of this talk wasn’t superb, so there might have been some incorrect facts above)

Keller – What do Quiver Mutations correspond to? He showed us neat facts about them, and told us that other people do care. Why do they care? What does a mutation correspond to? Also, he wrote down an algebra of Ginzburg’s associated to a quiver with superpotential. It was very close to the preprojective algebra of the quiver (I believe they coincide when the superpotential is zero). Is there anyway to deform this algebra, so that it corresponds to deformed preprojective algebras?

Lurie – I would have liked to see some examples of Koszul duality for algebras. In particular, he never mentioned what the Koszul dual of the motivating example, , was. Well, presumably we would need to make it an algebra first, but is almost certainly an coalgebra, and so has a dual. I believe someone asked a question to this effect, and Lurie answered ‘its just the delooping’, which doesn’t make much sense to me.

Me – If I have a Lie groupoid which is sufficiently nice, how do I construct a classifying space for it? I’m interested in equivariant cohomology for Lie groupoids. Is there a Weil algebra for a nice Lie groupoid?

Me – Is there a good resource for the metatheory of the ‘field with 1 element’? I’ve read several posts of John Baez on the subject, and I think its neat enough that I’d like to give a graduate colloquium on it. However, I’d like a more complete resource before I talk, and I haven’t found one.
I’m sure I had more questions, but I didn’t think to start writing them down til the fourth day. Anyway, thanks again to the speakers for a delightful conference.
March 19, 2008 at 12:28 pm 
Hi Greg. We chatted at the conference about quivery things.
On Okounkov: I don’t know much about even the classical CalogeroMoser system, but I’d like to learn more, so I’m starting to read Etingof’s notes arXiv:math/0606233, which also discuss the classical case. Certainly looks related to Okounkov’s business.
On Seidel: In the talk his symplectic manifolds were deformations of curve singularities, I think. If you do this for surface singularities, say type A_n, then you do in fact get the double of the affine Dynkin diagram of type A_n, quite naturally. The algebra is finite dimensional and Koszul, but the Koszul dual relations are not quite the same as the preprojective relations. Instead, it is generated by sums of loops out of each vertex. They generate the same ideal for n odd, but only something Morita equivalent for n even. Homological mirror symmetry is understood for the A case (on the coherent side you get the resolution of the singularity rather than the deformation) and should probably work for the D and E cases though I didn’t get to discuss this with Seidel as much as I would have liked. See the paper of Ishii, Ueda, and Uehara arXiv:math/0609551.
On Keller: One speaker that I was able to have a real conversation with. What I got out of this conversation:
1. Mutation really is a generalization of the BGP reflection functors, which occur at a source or a sink. Now we know what to do at a vertex that is neither a source nor a sink. One can ask what is the image of the standard heart of D(Q)
in D(Q’) under this generalized reflection functor. The answer: tilt at the simple where you are mutating, just like for BGP.
2. One of Ginzburg’s main examples is when a finite group G in SL(V) acts on V.
You’d like to understand Gsheaves on V or sheaves on GHIlb(V), when it is a resolution of the quotient V/G. In this example, if the quiver you take is the McKay quiver minus the vertex corresponding to the trivial rep, then the dg algebra coming from the superpotential describes the subcategory of the derived category of G sheaves where the trivial representation does not appear in the global sections. Keller told me that on the GHilb(V) side, this is the kernel of the pushforward along the HilbertChow morphism to V/G.
I hope that is of use to somebody. I can elaborate more if it would help.
March 20, 2008 at 8:58 am 
Hi – a couple of quick comments.
First about Costello’s remark about S^1 actions:
this is a purely homotopical S^1, and its actions are
quite different from the C^* action giving a grading.
The way it typically manifests itself – as in his talk –
is that an S^1 action on a space gives an action of
the homology of S^1 (the topologists’ form of the “group
algebra” of a topological group) acts on the cohomology
of X. The homology of S^1 has a single nonunit element, in
degree 1, hence the extra differential — this is how Connes’
cyclic differential on the Hochschild chain complex appears
naturally, since Hochschild chains in topology are functions
(or cochains) on the loop space, which carries the obvious
circle action..
(there’s more on this in Lurie’s beautiful Morse Lectures –
I have notes on my webpage).
Another quick note — the algebra Ginzburg assigns
to a quiver with potential does indeed look a lot
like a preprojective algebra but it has different grading —
ie it corresponds to something like a super form of the
cotangent bundle of the quiver variety, not the usual
cotangent bundle like the preprojective algebra.
As for Lurie, the basic example of Koszul duality
in topology is passing from based loops on a space to
the space itself, ie delooping — this is achieved concretely
by the bar construction. One can think of this
as the relation between a group G and its classifying space
BG — G is based loops on BG, and you recover BG
by taking the geometric realization of the
simplicial complex with simplices pt, G, GxG, GxGxG,…
If you look in the literature on Koszul duality for algebras
you’ll see the same formal structure appearing.
For example in the Koszul duality between exterior
and symmetric algebras, the exterior algebra
plays the role of G (it is the homology of a product of S^1’s)
and the symmetric algebra, the role of BG (it is the cohomology
of the corresponding classifying space).
March 20, 2008 at 12:10 pm 
A more important question, though: am I the only one who got a horrible cold/flu from this conference?
March 25, 2008 at 3:12 pm 
I think our coverage of the field with one element is largely confined to the following four posts, I,
II,
III, IV, the last of which points you to This Week’s Finds 259 and a short bibliography.