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Table of Contents 0. Introduction. 1. Operads in algebra and geometry. 2. Quadratic operads. 3. Duality for dg -operads. 4. Koszul operads. 0. Introduction. (0.1) The purpose of this paper is to relate two seemingly disparate developments. One is the theory of graph cohomology of Kontsevich [Kon 2 -3] which arose out of earlier works of Penner [Pe] and Kontsevich [Kon 1] on the cell decomposition and intersection theory on the moduli spaces of curves. The other is the theory of Koszul duality for quadratic associative algebras which was introduced by Priddy [Pr] and has found many applications in homological algebra, algebraic geometry and representation theory (see e.g., [Be] [BGG] [BGS] [Ka 1] [Man]). The unifying concept here is that of an operad.This paper can be divided into two parts consisting of chapters 1, 3 and 2, 4, respectively. The purpose of the first part is to establish a relationship between operads, moduli spaces of stable curves and graph complexes. To each operad we associate a collection of sheaves on moduli spaces. We introduce, in a natural way, the cobar complex of an operad and show that it is nothing but a (special case of the) graph complex, and that both constructions can be interpreted as the Verdier duality functor on sheaves.In the second part we introduce a class of operads, called quadratic, and introduce a distinguished subclass of Koszul operads. The main reason for introducing Koszul operads (and in fact for writing this paper) is that most of the operads "arising from nature" are Koszul, cf. (0.8) below. We define a natural duality on quadratic operads (which is 1

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Chapter 0. Chow quotients. 0.1. Chow varieties and Chow quotients. 0.2. Torus action on a projective space: secondary polytopes. 0.3. Structure of cycles from the Chow quotient. 0.4. Relation to Mumford quotients. 0.5. Hilbert quotients (the Bialynicki-Birula-Sommese construction). Chapter 1. Generalized Lie complexes. 1.1. Lie complexes and the Chow quotient of Grassmannian. 1.2. Chow strata and matroid decompositions of hypersimplex. 1.3. An example: matroid decompositions of the hypersimplex ∆(2, n). 1.4. Relation to the secondary variety for the product of two simplices. 1.5. Relation to Hilbert quotients. 1.6. The (hyper-) simplicial structure on the collection of G(k, n)//H.