Memoirs of the American Mathematical Society 2013; 116 pp; softcover Volume: 230 ISBN10: 0821892126 ISBN13: 9780821892121 List Price: US$75 Individual Members: US$45 Institutional Members: US$60 Order Code: MEMO/230/1079
 The little \(N\)disks operad, \(\mathcal B\), along with its variants, is an important tool in homotopy theory. It is defined in terms of configurations of disjoint \(N\)dimensional disks inside the standard unit disk in \(\mathbb{R}^N\) and it was initially conceived for detecting and understanding \(N\)fold loop spaces. Its many uses now stretch across a variety of disciplines including topology, algebra, and mathematical physics. In this paper, the authors develop the details of Kontsevich's proof of the formality of little \(N\)disks operad over the field of real numbers. More precisely, one can consider the singular chains \(\operatorname{C}_*(\mathcal B; \mathbb{R})\) on \(\mathcal B\) as well as the singular homology \(\operatorname{H}_*(\mathcal B; \mathbb{R})\) of \(\mathcal B\). These two objects are operads in the category of chain complexes. The formality then states that there is a zigzag of quasiisomorphisms connecting these two operads. The formality also in some sense holds in the category of commutative differential graded algebras. The authors additionally prove a relative version of the formality for the inclusion of the little \(m\)disks operad in the little \(N\)disks operad when \(N\geq2m+1\). Table of Contents  Introduction
 Notation, linear orders, weak partitions, and operads
 CDGA models for operads
 Real homotopy theory of semialgebraic sets
 The FultonMacPherson operad
 The CDGAs of admissible diagrams
 Cooperad structure on the spaces of (admissible) diagrams
 Equivalence of the cooperads \(\mathcal{D}\) and \(\mathrm {H}^*(\mathrm{C}[\bullet])\)
 The Kontsevich configuration space integrals
 Proofs of the formality theorems
 Index of notation
 Bibliography
