About a week ago (of November 22-23) I was in Riverside, California for Groupoidfest ’08. (Slides for the talk I gave here, and also in pdf.) It’s taken me a while to write it up, because I’ve been, among other things, applying for jobs.
This would be the second time I’ve been to Groupoidfest, and the first time I’ve been back in Riverside since I graduated from UCR last summer. While I was there, I also had the chance to talk to John Baez and some of his other students, past and present, and also to attend Alan Weinstein’s colloquium talk on Friday. On top of that, I managed to see a couple of my other friends in town, so all in all, it was a good trip.
There were quite a few talks, several of which were fairly short, so I’ll comment on a few examples which I found particularly relevant to me. So for instance Alan Paterson’s talk on Equivariant K-Theory for Proper Groupoids: here’s a case where I’m seeing familiar issues from a different direction. K-theory studies objects by looking at categories of vector bundles. Equivariant K-theory can be taken to mean that these bundles come with isomorphisms between fibres which come from a group action, or more generally the morphisms of a groupoid. It’s a kind of categorification of equivariant cohomology. Alan Paterson’s talk was quite extensive, but there’s a whole vocabulary here I’m still learning. The culmination of the talk dealt with Hilbert bundles (a little more structured than vector bundles), and the the Hilbert bundle (where is the space of morphisms of a groupoid – so this can be treated as a bundle over the space of objects induced by, say, the map taking a morphism to its target object). This bundle has the nice “stabilization” property that taking a direct sum with any other bundle leaves it unchanged.
John Quigg also spoke about Hilbert bundles, and “Fell bundles” (he spoke about these last year, too), but since John Baez described this in more detail in his report on GFest, I’ll just remark on another aspect of this talk, where he was using a “disintegration theorem”, which was more familiar to me. This says that every representation of the convolution algebra of a groupoid comes from direct-integrating some Hilbert bundle. This is reminiscent of the decomposition of any von Neumann algebra as a direct integral of “factors” (which are each subalgebras of the algebra of operators on fibres of some Hilbert bundle). There seem to be a lot of these “disintegration theorems” involving direct integrals. I have some ideas about this, but I’ll hold off on them until they’re a little more developed.
There were a number of other talks with interesting elements, but many were a bit too short for me to get much more than an awareness that there’s interesting work being done that I’d like to learn more about: Xiang Tang’s talk on “Group Extensions and Duality of Gerbes” seemed to be perhaps related to what I would describe as 2-vector spaces generated by -groupoids, but using (blush) a more standard language; Joris Vankerschaner talked about classical mechanics on Lie Groupoids – in particular, discrete field theories valued in groupoids.
Now, the colloqium talk by Alan Weinstein was titled “Groupoid Symmetry for Einstein’s Equation?”, including the querulous punctuation, since some of it was speculative. The basic idea behind this talk was to apply groupoids to General Relativity, thought of as an evolution equation. The Hamiltonian formulation of GR describes a spacelike hypersurface evolving in time – this was described by Arnowitt, Deser and Misner, or ADM, from whom we likewise get the “ADM mass”, which can be thought of as the energy of the worldsheet, as it’s seen by an observer at spacelike infinity. This formulation doesn’t describe all solutions of Einstein’s equations – in particular, nothing with closed timelike curves, and unless I misremember, really only makes sense for asymptotically flat spacetimes – certainly that’s true for the ADM mass. But it does fit with our usual intuitions about systems evolving in time, and makes some initial-value problems – including local ones – more or less tractable, which is good for practical purposes. (There are still further technical provisos to ensure the result actually satisfies Einstein’s equations.)
(Note: on looking the asymptotic flatness issue up in Wald’s book, it seems that even for compact space slices, although it naively appears the Hamiltonian vanishes, this can possibly be resolved by some tricky “deparametrization” Wald doesn’t entirely explain. The restriction against closed timelike curves alone probably won’t dissuade anyone who isn’t dead set on building a time machine.)
Anyway, the groupoid symmetry Weinstein was suggesting involves taking space slices (or some slight variation thereon, such as thickened slices, or slices equipped with a metric) to be objects, and considering diffeomorphisms between slices as morphisms. This would make a Lie groupoid, and the corresponding Lie algebroid would reproduce an algebroid structure which it’s natural to associate with the phase space for the system (specifically, one associated with the Poisson structure. There are some links on this correspondence over on the n-Category Cafe – basically, a Poisson algebra is like a Lie bracket structure on the tangent Lie algebroid to a manifold).
Where one can go with this idea, I’m not sure. More clear to me, since I’ve thought about it more, was the content of Weinstein’s other talk – about the volume of a differentiable stack – which he gave at the conference proper. This is a smooth/differentiable version of groupoid cardinality, and therefore has all sorts of applications to groupoidification in both the vector-space and 2-vector-space flavours. The basic point is that groupoid cardinality – for finite groupoids – involves measures in two ways. One way to find it is as a sum over all the objects; the sum is of the quantities , the reciprocals of the numbers of morphisms ending at object . There are other, equivalent, ways, but they all amount to sums of reciprocals of numbers found by counting. Both these numbers, and the sums themselves, can be seen as integrals – using the counting measure on a discrete, finite set. The first point of the talk was that counting measure should be replaced by two kinds of measure – one on the object spaces, and one for morphisms – when one passes to a differentiable stack.
(There are a variety of ways to think about stacks – one is that a stack is a groupoid, thought of up to equivalence. In which case, the real information in a stack consists of the set of isomorphism classes of object, or orbits, and also the automorphism, or isotropy, groups for objects in each orbit. One good thing about stacks is that they keep track of information which is lost when taking quotients – if a point is fixed under a group action, for instance, it still has nontrivial isotropy when taking the “stacky” quotient of a set by a group action. A nice representative groupoid for this quotient is keeps all points, but adds morphisms corresponding to motions under the action.)
There were also a bunch of talks about groupoidification of linear algebra, by John Baez and his current students – about groupoidification of linear algebra. Since I’ve written about this a lot here anyway, I’ll just remark that Christopher Walker introduced the concept, Alex Hoffnung talked about applications to Hecke algebras and incidence geometries (also discussed in their seminar starting here), while John spoke about Jim Dolan’s ideas for groupoidifying the harmonic oscillator, which have also been written up and slightly expanded by me. My own talk is also sort of about groupoidification, albeit a higher-dimensional version thereof.
At any rate, that’s about all I have time to say about GFest ’08, although there were many other talks which reinforced my desire to keep learning more about all the wonderful stuff known to people who study groupoids, and especially Lie groupoids.