Hi guys, sorry it took me so long to decide to post something. I was at the Chicago conference and I agree that it was a very interesting event. Unfortunately, I did not go to all the talks (I am a bit overloaded with conferences as the moment, so - much as I would like it - I never manage to attend more than a couple of talks a day), so I can only give a very partial overview of what was happening there. One talk I very much regretted missing was the one Sasha Goncharov gave, which everyone said was great. If someone would be so kind as to give me a quick summary of it, I'd love to hear more about it. Another one I would have been really interested in but I also missed due to having miscalculated the time (shame on me) was the one Kevin Costello gave on renormalization. In both cases, I very much hope that papers will be soon available on the archive (keep an eye on that). I missed many other talks I would have really liked to attend, which I will not try to list here.
Now for something I did see: David Ben-Zvi's talk was very inspiring. It was based on the joint work with David Nadler, who gave one more talk on the subject. The paper is on the archive at arXiv:0706.0322. The main idea is to use loop space geometry to build a parallel between the category of equivariant D-modules on flag varieties associated to reductive groups and categories of equivariant coherent sheaves on Steinberg varieties, via a version of S^1 localization. I am not doing justice to this really interesting work with these few words, but please do look at the paper on the eprints, and since you are at it take a look also at the work of Ben-Zvi and Thomas Nevins (also a speaker at the conference and a former classmate of mine in graduate school).
Yuri Manin gave the introductory talk of the workshop on his work with Dennis Borisov (math/0609748) on internal cohomomorphisms for operads. A broad context envisioned for dealing with symmetries and moduli problems in noncommutative geometry. Starting with internal cohomomorphisms of associative algebras, which was the way Manin approached quantum group symmetries of noncommutative spaces in his noncommutative geometry and his quantum groups books. From that starting point the theory is developed to include the kind of operadic constructions (as functors of labelled graphs) that are essential in the theory of moduli spaces of curves. This is an operadic version of noncommutative geometry designed to carry over moduli problems and combine them with Hopf algebra symmetries.
Another talk I enjoyed was the one given by Spencer Bloch. That's work in progress, so once again keep an eye on upcoming stuff (probably on Bloch's webpage if not on the archive). His current work is related to "Feynman motives" namely motives associated to graph hypersurfaces, that are meant to realize Feynman integrals as periods. On this you can look at the very nice Takagi Lectures on Bloch's webpage as well as the famous Bloch-Esnault-Kreimer paper . In the Chicago talk he described the role of certain compactifications due to Betsvina-Feighn of Out(F_n) in studying resolutions of singularities for graph hypersurfaces and the relation between the Connes-Kreimer Hopf algebra and Kontsevich's graph homology, as well as on the lifting of the CK Hopf algebra at the motivic level, a theme already discussed in his Takagi Lectures.
Jonathan Block gave a talk on a derived categories framework for spectral triples in terms of curves DGA's. This looks like a very promising viewpoint, especially in applications of spectral triple to algebro-geometric or number-theoretic contexts. This framework is the basis of an upcoming series of papers by the author, of which the first two are available on the archive as math/0509284 and math/0604296. These are meant to provide a setting where Mukai transform and Baum-Connes conjecture naturally interact.
Another really nice talk I attended was that given by Sasha Polishchuk on solutions to the associative Yang-Baxter equations (and relations between these and quantum and classical Yang-Baxter equations) obtained from elliptic curves and degenerations thereof. You can find some of this in the paper math/0612761.
Masha Vlasenko gave a nice talk on her recent number theoretic work on the "Eisenstein cocycle" (see math/0611214). Her previous work on theta functions of noncommutative tori with real multiplication is very interesting too (see math/0601405).
Among other talks I attended there was a nice one by Voronov on duality in graph homology, which gives a very nice identification between Koszul duality for operads and Verdier duality for constructible sheaves on spaces of graphs, in particular for the case of "Outer space" X_n/Out(F_n) with X_n the space of metric graphs with markings and the case of moduli spaces of curves, realized via moduli spaces of ribbon graphs.
The talk was based on joint work of Voronov with Lazarev available as math/0702313.
Victor Ginzburg talked about Calabi-Yau algebras (see math/0612139) in relation to Kai Behrend's perfect obstruction theory (related to Behrend's nice paper on Donaldson-Thomas invariants math/0507523) and the relation of CY algebras to quiver representations. There were some nice examples like Heegaard splittings of 3-manifolds, quantum del Pezzo surfaces and McKay correspondence in dimension 3.
I won't comment on the other talks. In fact, I apologize to all the speakers I mentioned here for the inaccuracies and outright mistakes I made in reporting on their talks. I also apologize to all the speakers whose talks I missed or attended but did not mention here. I just wanted to give a brief feeling of the general flavor of the conference. It would be very nice if other people who attended it would like to complement this very partial report with something more accurate.