Abstract. We construct a solution of the master equation by means of standard tools from homological perturbation theory under just the hypothesis that the ground field be of characteristic zero, thereby avoiding the formality assumption of the relevant dg Lie algebra. To this end, we endow the homology H(g) of any differential graded Lie algebra g over a field of characteristic zero with an sh-Lie structure such that g and H(g) are sh-equivalent. We discuss our solution of the master equation in the context of deformation theory. Given the extra structure appropriate to the extended moduli space of complex structures on a Calabi-Yau manifold, the known solutions result as a special case.

Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the Lie triple systems kind. A basic tool is the Maurer-Cartan algebra—the algebra of alternating forms on a vector space so that Lie brackets correspond to square zero derivations of this algebra—and multialgebra generalizations thereof. The higher homotopies are phrased in terms of these multialgebras. Applications to foliations are discussed: objects which serve as replacements for the Lie algebra of vector fields on the “space of leaves ” and for the algebra of multivector fields are developed, and the spectral sequence of a foliation is shown to arise as a special case of a more general spectral sequence including as well the Hodge-de Rham spectral sequence.

Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940’s. Prompted by the failure of the Alexander-Whitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor’s classifying space construction to associative H-spaces, and a careful examination of this extension led Stasheff to the discovery of An-spaces and A∞-spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950’s, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic

To the memory of G. Chogoshvili Abstract. Let R be a local ring and A a connected differential graded algebra over R which is free as a graded R-module. Using homological perturbation theory techniques, we construct a minimal free multi model for A having properties similar to that of an ordinary minimal model over a field; in particular the model is unique up to isomorphism of multialgebras. The attribute ‘multi ’ refers to the category of multicomplexes. 2000 Mathematics Subject Classification. 18G10, 18G35, 18G55, 55P35, 55P62, 55U15, 57T30. Key words and phrases. Models for differential graded algebras, minimal models for differential graded algebras over local rings, multicomplex, multialgebra, homological perturbations. 2 JOHANNES HUEBSCHMANN

Dedicated to Alan Weinstein on his 60’th birthday Abstract. Higher homotopies generalizations of Lie-Rinehart algebras, Gerstenhaber-, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the Lie triple systems kind. A basic tool is the Maurer-Cartan algebra—the algebra of alternating forms on a vector space so that Lie brackets correspond to square zero derivations of this algebra—and multi algebra generalizations thereof, and the higher homotopies are phrased in terms of these multi algebras. Applications to foliations are discussed, including objects which serve as a replacement for the Lie algebra of vector fields on the space of leaves and for the algebra of multi vector fields, as well as a construction of the spectral sequence of a foliation. 2000 Mathematics Subject Classification. Primary 17B65 17B66 53C12 57R30; secondary 17B55 17B56 17B81 18G40 53C15 55R20 70H45. Key words and phrases. Quasi-Lie-Rinehart algebra, quasi-Gerstenhaber algebra, quasi-Batalin-Vilkovisky algebra, Lie-Rinehart triple, Maurer-Cartan algebra, reductive homogeneous space, foliation, higher homotopies, Jacobi identity up to higher homotopies, spectral sequence of a foliation. 2 JOHANNES HUEBSCHMANN

In honor of the 60-th birthday of Tornike Kadeishvilli Early in the history of higher homotopy algebra [Sta63], it was realized that Massey products are homotopy invariants in a special sense, but it was the work of Tornike Kadeisvili that showed they were but a shadow of an A∞-structure on the homology of a differential graded algebra. Here we relate his work to that of Victor Gugenheim [Gug82] and K.T. (Chester) Chen [Che73a]. This paper is a personal tribute to Tornike and the Georgian school of homotopy theory as well as to Gugenheim and Chen, who unfortunately are not with us to appreciate this convergence.