Shuffle operads and their applications

The notion of a symmetric operad is a generalisation of that of an associative algebra, just for a monoidal category larger than the tensor category of vector spaces. For a long while, several ingredients that proved to be useful in homological algebra for associative algebras (e.g. Groebner bases, Anick-type resolutions etc.), have been missing in the case of operads, since the monoidal category in question appears to have too rich structure for those approaches to work. In this talk, I shall discuss shuffle operads, as defined by myself in a joint work with A.Khoroshkin, which appear to allow many approaches valid for associative algebras to be adapted for the operadic case. Every symmetric operad can be considered as a shuffle operad, and that shuffle operad keeps track of many homological properties of the original symmetric operad. Among the applications of this approach, I shall talk about Koszul duality for operads, bar homology for the Batalin-Vilkovisky operad and the Rota-Baxter operad, and combinatorics of pattern avoidance.