This is a record of the research seminar we run at University of Copenhagen in 2013.
September 25: Makoto talked about the comparison of cup products in group cohomology and cyclic cohomology of group algebra.
September 6: Markus continued about the topological / categorical computation of the (homotopy) center of simplicial categories. Note by Makoto.
August 6: Makoto gave an expository talk on “Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields” arXiv:1211.4230 by Dolgushev, Rogers, and Willwacher. Note by Makoto.
July 2: Angela talked about the natural operations on Hochschild chains of commutative Frobenius algebras. Her note.
June 24: Kristian talked about the cyclic and Hochschild groups of the group rings.
June 19: Markus talked about the topological / categorical approach to the HH0. Note taken by Makoto.
May 29: Nathalie talked about the diagramatic proof of the Cartan homotopy formula. Note taken by Makoto. Next seminar is June 19 or 20.
May 2: Makoto talked about the application of Cartan caluculus to the noncommutative torus. Note taken by Makoto.
April 25: Makoto talked about the Cartan calculus on the cyclic complex based on Getzler’s paper. Note by Makoto.
April 11: Rasmus talked about the X-complex approach to the noncummutative torus based on Nekljudova’s thesis. The next seminar will be at Thursday 25 April 10:15 AM (04.4.1), about the noncommutative Cartan calculus by Makoto.
April 4: Tyrone talked about the X-complex formalism of Cuntz-Quillen. Note taken by Makoto. The next seminar will be at Thursday 11 April 1:00 PM, about the X-complex of the NC-torus by Rasmus.
March 15: Ryszard gave an introductory talk. Note taken by Makoto. The next seminar will be on the first week of April. We will start from different calculations of cyclic (co)homology for the noncommutative torus.
Early March: Below is a quick summary of the subjects I compiled for brainstorming, and initially circualted as an email to peers.
As a start, below is a short summary of the (algebraic) Hochschild / cyclic (co)homology theories. It’s done in hurry and no wonder heavily biased to my preference, so feel free to add your input. I tried to indicate the references at the bottom, but I appreciate if you could expand that as well.
You could say that there are two large camps studying the Hochschild / cyclic (co)homology theories: the noncommutative differential geometry (NCDG) a la Connes, Cuntz-Quillen…, and the noncommutative calculus (NCC) a la Tsygan, Kontsevich…
The emphasis is more on the relation with the K-groups. So, the periodic cyclic groups HP∗(A) and HP∗(A) are very important. On the Hochschild side, the Hochschild homology complex C∗(A, A) with coefficient A, and its dual complex, the Hochschild cohomology complex C∗(A, A∗) with coefficient A∗. These correspond to the differential forms and currents, which are receptacles of the “Chern characters”.
Applications are mainly to foliations and index theory things, which are approachable from the operator algebras. It often becomes a problem to find a suitable algebra which is large enough to K-theoretically coincide with the C*-algebra, at the same time small enough to support interesting cocycles. Connes’s “transverse fundamental class” paper is a tour de force containing many important ideas in this regard. The local index formula for spectral triples (Connes-Moscovici, Higson’s survey is more accessible) can be regarded as an variation of the Atiyah-Singer index formula in this context. I should mention application to the Novikov conjecture by Connes-Moscovici. Boris also gave a SYM lecture about it some time ago.
Systematic computation is done for commutative algebras (Hochschild-Kostant-Rosenberg, Connes) and group algebras (Karoubi, Burghelea; in Loday’s book), to name a few.
From the viewpoint of relativising the index theory, there’s the bivariant theory of Cuntz-Quillen. This has some relation to the KK-theory / E-theory also.
Compared to the NCC, you often want to work with non-formal deformation.
The references are Loday’s old book (more on the homology side, also with topological viewpoint) and Cuntz’s surveys (also on the cohomology side, plus bivariant things).
The emphasis is more on the Hochschild cohomology complex C∗(A, A) with the coefficient A, which is a generalization of the polyvector fields. Deformation quantization is the principal target, so the comparison (‘formality’) to more geometric constructions such as the Poisson (co)homology is important. When the deformation comes into play, it’s mostly formal. This field is littered by so many conjectures (which is both good and bad).
Cartan calculus is an important formalism here. First, there was the Cartan homotopy formula by Getzler, later by Nest-Tsygan, Daletsky-Tsygan, Tamarkin-Tsygan, etc. The (homotopy) Batalin-Vilkovisky algebra is the keyword. This leads to the Gauss-Manin connection on periodic cyclic theory. Capturing Connes’s B-differential was somewhat (still ongoing?) problem here. The introduction of Tamarkin-Tsygan is probably a good entry point.
For the sake of formality, the operadic approach is also important here. There’s Loday-Vallette’s book about the algebraic approach to operads in general. Kontsevich-Soibelman have several surveys with many conjectures, but some of them might be getting outdated (in terms of what is proved and what is not). Hinich’s “Tamarkin’s proof of Kontsevich formality” might be a good motivating start. And Dolgushev-Tamarkin-Tsygan.
For deformation quantization, there’s Dolgushev’s work on Van den Bergh duality, which is a nice application of NCC to NCDG.
There’s Nest-Tsygan’s algebraic index theorem.
The Gauss-Manin connection should be one such. Also, there are several things first proved in the formal deformation setting and then brought to the strict deformation setting. Dolgushev’s Van den Bergh duality theorem.