Bonn-Luxembourg-Strasbourg Days:

Title : Gerstenhaber-Schack diagram cohomology from the operadic point of view

Abstract : Gerstenhaber and Schack defined a cohomology theory for diagrams of associative algebras. We consider the coloured operad $A$ describing such diagrams and our aim is to construct operadic cohomology for $A$-algebras. Therefore we would like to find a free resolution of the coloured operad $A$. This problem is already well understood e.g. for a single morphism (of algebras over Koszul operads). But for more complicated diagrams, little is known. We first state a theorem on existence of such a resolution. It is partially explicit but not to the extend so that we are able to write down the cohomology for $A$-algebras. Then we show that the operadic cohomology can be generally expressed as an Ext functor in the category of operadic modules. Therefore it suffices to construct a free resolution of a specific operadic module. As an application, we prove that Gerstenhaber-Schack diagram cohomology is indeed operadic cohomology.

Title : Curved algebras and curved graph complexes

Abstract: Let O be a cyclic, or, more generally, modular, operad and consider its Feynman transform F(O). Without the usual stability conditions (which, for example, prevent O from being unital) the operad F(O) is rather different from what one is used to; in particular it is usually acyclic in all arities save the bottom one (corresponding to the graphs with no legs). The corresponding legless part, however, could have nontrivial homology and we compute it in the cases when O is the operad governing unital associative or unital commutative algebras. This amounts to computing the homology of commutative and ribbon graph complexes when one allows vertices of valences one and two. The algebras over F(O) can sometimes be interptered as curved infinity algebras and we discuss the problem of classifying such algebras. A complete answer could be given in the case of nontrivially curved A-infinity or L-infinity algebras.

Title : "Hidden structures in homological algebra"

Abstract : If a chain complex is equipped with some compatible algebraic structure, then its homology gets equipped with this algebraic structure. The surprise is that, in most cases, there is a hidden algebraic structure on this homology. We will give several elementary examples and show how the notions of spectral sequence, Connes boundary map B, A-infinity algebra and MacLane invariant of a crossed module come naturally out of this principle. I'll end up with a problem in biology.

Title : Koszul complex = cotangent complex

Abstract: We extend the Koszul duality theory of associative algebras to any type of algebras, or more precisely, to algebras over an operad. Recall that in the classical case, this Koszul duality theory relies on an important chain complex: the Koszul complex. We show that the cotangent complex, involved in the cohomology theory of algebras over an operad, generalizes the Koszul complex.

Title : Homotopy theory of homotopy properads

Abstract : Homotopy properads generalize properads the way A-infinity algebras generalize associative algebras. In this talk we will begin with discussing transfer of homotopy properad structures. From this one can show that quasi-isomorphisms are invertible. We will then discuss implications of this on the homotopy category of homotopy properads.

Title : Homotopy Batalin-Vilkovisky algebras