Operads 2009

An operad is a mathematical device which allows one to encode universally a wide variety of algebraic structures. Unraveled in the seventies in algebraic topology to recognize n-fold loop spaces, it has enjoyed a renaissance in the nineties mainly under the impulse of quantum field theories. This universal notion is now used in many domains of mathematics like differential geometry (deformation theory), algebraic geometry (moduli spaces of curves, Gromov-Witten invariants), noncommutative geometry (cyclic homology), algebraic combinatorics (Hopf algebras), theoretical physics (field theorics, renormalization), computer science (rewriting Systems) and universal algebra. The purpose of the present volume is to présent contributions of the notion of operad in these fields, where it plays an important rôle. It follows a school and a conference « Operads 2009 » which took place at the CIRM (Luminy, France) in April 2009.