NWDR Workshop Summer 2012 – Abstracts
Hanno Becker (Bonn)
Models for Singularity Categories
This talk is about the construction of various Quillen models for categories of singularities. I will begin by recalling the connection between abelian model structures, cotorsion pairs and deconstructible classes and then describe the construction of the singular models. Next, I will explain how Krause's recollement for the stable derived category can be obtained model categorically. Finally, as an example I will show that Positselski's contraderived model for the homotopy category of matrix factorizations is Quillen equivalent to a particular singular model structure on the category of curved mixed complexes.
William Crawley-Boevey (Leeds)
D-modules for nodal curves and multiplicative preprojective algebras
In work on the Deligne-Simpson problem, I introduced (with Shaw), certain algebras, called multiplicative preprojective algebras, and I also studied monodromy of logarithmic connections on vector bundlles on the Riemann sphere.
In order to directly connect these notions, I showed in arXiv:1109.2018 [math.RA], that multiplicative preprojective algebras provide a natural receptacle for monodromy for certain systems of vector bundles, linear maps and logarithmic connections on what I called 'Riemann surface quivers'.
Instead of this perhaps artificial notion, I was asked whether or not it could be formulated in terms of torsion-free sheaves on nodal curves. This talk is a response to that question. I will discuss various categories of D-modules on nodal curves, and then a modification which is related to multiplicative preprojective algebras.
Andrew Hubery (Leeds)
Braid group actions and presentations of affine quantum groups
We investigate the extent to which the affine quantum group is determined by the two properties of containing the corresponding quantum group of finite type and having an action of the (extended) affine braid group. Using a presentation of the braid group (analogous to the presentation of the extended affine Weyl group as the semi-direct product of the finite Weyl group and the weight lattice), we see that this is well adapted to Drinfeld's new presentation of the quantum group and thus obtain a conceptual proof of the isomorphism between this and the Drinfeld-Jimbo presentation.
Ulrich Krähmer (Glasgow)
Batalin-Vilkovisky Structures on Ext and Tor
The topic of this talk (based on joint work with Niels Kowalzig) is an algebraic structure whose best known example is provided by the multivector fields and the differential forms on a smooth manifold: the multivector fields are a Gerstenhaber algebra with respect to wedge product and Schouten-Nijenhuis bracket, and the differential forms are what we call a Batalin-Vilkovisky module over this Gerstenhaber algebra, which means that the multivector fields act in two ways on forms – by means of contraction and of Lie derivative – and that these actions are related by a differential that fits into Cartan's "magic" homotopy formula. Nest, Tamarkin and Tsygan suggested to refer to this abstract package of two graded vector spaces with such operations as to a noncommutative differential calculus.
Generalising work by the above mentioned authors and by Getzler, Gerstenhaber, Goodwillie, Huebschmann, Rinehart and others, I will explain that Ext and Tor over Hopf algebroids tends to carry such a structure which means that homological algebra produces plenty of examples of noncommutative differential calculi, including for example Hochschild and Poisson (co)homology. As a first application, Ginzburg's theorem that the Hochschild cohomology ring of a Calabi-Yau algebra is a Batalin-Vilkovisky algebra is extended to twisted Calabi-Yau algebras such as quanum groups, quantum homogeneous spaces or quantum vector spaces.
Antoine Touzé (Paris)
Finite generation of the cohomology of reductive group schemes
It has been proved by Haboush that if G is a reductive group scheme acting on an algebra A commutative and finitely generated, then the invariants AG form a finitely generated algebra (this was conjectured by Mumford). Evens (1964), and later Friedlander Suslin (1997), proved similar finite generation theorems for the cohomology ring H*(G,A) for finite groups and finite group schemes G acting on a commutative finitely generated algebra A. After this, van der Kallen conjectured that all the reductive group schemes have finitely generated cohomology algebras.
In this talk we will present an overview of the proof of this theorem and how strict polynomial functors come into play for this problem.
NB: The finite generation result depends (directly or indirectly) on the contribution of many authors: Friedlander, Suslin, van der Kallen, Srinivas, Grosshans and Touzé, and a complete proof is available in the article: "Bifunctor cohomology and Cohomological finite generation for reductive groups" Duke Math. J. 151 (2010).