Talks
Saturday 15
Saturday 15
Pierre Schapira
09:30 - 10:30A finiteness theorem for holonomic DQ-modules on Poisson manifolds (joint work with Masaki Kashiwara)
On a complex symplectic manifold we prove a finiteness result for the global sections of solutions of holonomic DQ-modules in two cases: (a) by assuming that there exists a Poisson compactification (b) in the algebraic case. This extends our previous results in which the symplectic manifold was compact. The main tool is a finiteness theorem for R-constructible sheaves on a real analytic manifold in a non proper situation.
Rishi Vyas
11:00 - 12:00A noncommutative Matlis-Greenlees-May equivalence
The notion of a weakly proregular sequence in a commutative ring was first formally introduced by Alonso-Jeremias-Lipman (though the property that it formalizes was already known to Grothendieck), and further studied by Schenzel and Porta-Shaul-Yekutieli: a precise definition of this notion will be given during the talk. An ideal in a commutative ring is called weakly proregular if it has a weakly proregular generating set. Every ideal in a commutative noetherian ring is weakly proregular. It turns out that weak proregularity is the appropriate context for the Matlis-Greenlees-May (MGM) equivalence: given a weakly proregular ideal I in a commutative ring A, there is an equivalence of triangulated categories (given in one direction by derived local cohomology and in the other by derived completion at I) between cohomologically I-torsion (i.e. complexes with I-torsion cohomology) and cohomologically I-complete complexes in the derived category of A. In this talk, we will give a categorical characterization of weak proregularity: this characterization then serves as the foundation for a noncommutative generalisation of this notion. As a consequence, we will arrive at a noncommutative variant of the MGM equivalence. This work is joint with Amnon Yekutieli.
Bernhard Keller
14:30 - 15:30Tate-Hochschild cohomology from the singularity category
The singularity category (or stable derived category) was introduced by Buchweitz in 1986 and rediscovered in a geometric context by Orlov in 2003. It measures the failure of regularity of an algebra or scheme. Following Buchweitz, one defines the Tate-Hochschild cohomology of an algebra as the Yoneda algebra of the identity bimodule in the singularity category of bimodules. In recent work, Zhengfang Wang has shown that Tate-Hochschild cohomology is endowed with the same rich structure as classical Hochschild cohomology: a Gerstenhaber bracket in cohomology and a B-infinity structure at the cochain level. This suggests that Tate-Hochschild cohomology might be isomorphic to the classical Hochschild cohomology of a (differential graded) category, in analogy with a theorem of Lowen-Van den Bergh in the classical case. We show that indeed, at least as a graded algebra, Tate-Hochschild cohomology is the classical Hochschild cohomology of the singularity category with its canonical dg enhancement. In joint work with Zheng Hua, we have applied this to prove a weakened version of a conjecture by Donovan-Wemyss on the reconstruction of a (complete, local, compound Du Val) singularity from its contraction algebra, i.e. the algebra representing the non commutative deformations of the exceptional fiber of a resolution.
Michel Van den Bergh
16:00 - 17:00Comparing commutative and noncommutative resolutions of singularities
We report on joint work with Špela Špenko. Let a reductive group G act on a smooth variety X such that a good quotient X//G exists. We show that the derived category of suitable noncommutative crepant resolutions of X//G, can be embedded in the derived category of the Kirwan resolution of X//G. In fact, the embedding can be completed to a semi-orthogonal decomposition in which the other parts are all derived categories of Azumaya algebras over smooth Deligne-Mumford stacks.
Sunday 16
Sunday 16
Maxim Kontsevich
09:30 - 10:30Introduction to pre-Calabi-Yau structures
I will explain basic definitions of theory of pre-Calabi-Yau structures developed several years ago by Y.Vlassopoulos and myself. In a nutshell, pre-Calabi-yau structure on a small dg category gives a shifted Poisson structure on the moduli stack of objects given by the trace of a non-commutative polyvector field. For example, the category of constructible sheaves on an oriented stratified manifold is of such kind.
Jun-Ichi Miyachi
11:00 - 12:00On N-differential graded categories
We talk about the category of N-differential modules over an N-differential graded category, and describe the properties of its homotopy category and its derived category.
Reinhold Hübl
14:30 - 15:30Applications of Residues to Coding Theory
It is well known that divisors on a smooth curve C over a finite field k = $\mathbb{F}_q$ can be used to construct very powerful linear codes (Goppa–codes). We will use Amnon Yekutieli’s adelic approach to residues and residue complexes to obtain algebraic–geometric codes for general varieties X/k (of higher dimension and not necessarily smooth), generalizing Goppa–codes, we will study the parameters of these codes and we will compare the construction to other constructions of codes.
James Zhang
16:00 - 17:00Residue complexes and the Brown-Goodearl conjecture.
We use a noncommutative version of residue complexes introduced by Yekutieli to show that every weak Hopf algebra that is a finitely generated module over its affine center has finite self-injective dimension. Therefore the Brown-Goodearl conjecture holds in this special weak Hopf setting.
Here you can find a printable version of the schedule.