Operads 2009

An operad is a mathematical device used to encode universally a wide variety of algebraic structures. The name operad appeared first in the 1970s in algebraic topology to recognize $n$-fold loop spaces. Operads 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 such as differential geometry (deformation theory), algebraic geometry (moduli spaces of curves, Gromov-Witten invariants), noncommutative geometry (cyclic homology), algebraic combinatorics (Hopf algebras), theoretical physics (field theories, renormalization), computer science (rewriting systems) and universal algebra. The purpose of this volume is to present contributions about the notion of operads in these fields, where they play an important role. This volume is a result of a school and a conference, ''Operads 2009'', both of which took place at the CIRM (Luminy, France) in April 2009.