Dedicated to H. Keller on the occasion of his seventy fifth birthday Abstract. These are expanded notes of four introductory talks on A∞-algebras,

It is shown that any compact Kähler manifold M gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differential forms is again harmonic. If M happens to be a Calabi-Yau manifold, there exists a third strongly homotopy algebra closely related to the Barannikov-Kontsevich extended moduli space of complex structures. 1

Abstract. We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential “cup product ” on this sequence, satisfying a Leibniz rule up to curvature terms. It is not associative, but is part of an A∞-algebra of multilinear differential operators, which we also obtain explicitly. We illustrate the construction in the case of conformal differential geometry, where the cup product provides a wide-reaching generalization of helicity raising and lowering for conformally invariant field equations.

Abstract. Let A be a connected graded algebra and let E denote its Extalgebra i Exti A (kA, kA). There is a natural A∞-structure on E, and we prove that this structure is mainly determined by the relations of A. In particular, the coefficients of the A∞-products mn restricted to the tensor powers of Ext1 A (kA, kA) give the coefficients of the relations of A. We also relate the mn’s to Massey products.

We give a brief introduction to A1-algebras and show three contexts in which they appear in representation theory: the study of Yoneda algebras and Koszulity, the description of categories of ltered modules and the description of triangulated categories. Contents 1. Denitions, the bar construction, the minimality theorem 1 2. Yoneda algebras, Koszulity and ltered modules 5 3. Description of triangulated categories 8 References 10 1. Definitions, the bar construction, the minimality theorem 1.1. A-innity algebras and morphisms. We refer to [11] for a list of references and a topological motivation for the following denition: Let k be a eld. An A1 - algebra over k is a Z-graded vector space A = M p2Z A p endowed with graded maps (=homogeneous k-linear maps) mn : A

Abstract. We propose a new point of view of the Spencer cohomology appearing in the formal theory of differential equations based on a dual approach via comodules. It allows us to relate the Spencer cohomology with standard constructions in homological algebra and, in particular, to express it as a Cotor. We discuss concrete methods for its construction based on homological perturbation theory. Appears in Georgian Math. J., vol. 9, No. 4, 2002, 723-772. 1.

. A uniform and constructive approach for the computation of resolutions and for (co)homology computations for any nite p-group is detailed. The resolutions we construct ([32]) are, as vector spaces, as small as the minimal resolution of IFp over the elementary abelian p-group of the same order as the group under study. Our implementations are based on the development of sophisticated algebraic data structures. Applications to calculating functional cocycles are given and the possibility of constructing interesting codes using such methods is presented. 1 Introduction In this paper, we present a uniform constructive approach to calculating relatively small resolutions over nite p-groups. The algorithm we use comes from [32, 8.1.8 and the penultimate paragraph of 9.4]. There has been a massive amount of work done on the structure of p-groups since the beginning of group theory. A good introduction is [22]. We combine mathematical and computer methods to construct the uniform resolut...

A uniform and constructive approach for the computation of resolutions and for (co)homology computations for any nite p-group is detailed. The resolutions we construct ([29]) are, as vector spaces, as small as the minimal resolution of IFp over the elementary abelian p-group of the same order as the group under study. Our implementations are based on the development of sophisticated algebraic data structures. Applications to calculating functional cocycles are given and the possibility of constructing interesting codes using such methods is presented. 1

Computation of A ∞ algebras in group cohomology Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.)

Abstract. Kadeishvili’s proof of the minimality theorem (Kadeishvili, T.; On the homology theory of fiber spaces, Russian Math. Surveys 35 (3) 1980) induces an algorithm for the inductive computation of an A∞-algebra structure on the homology of a dg-algebra. In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete A∞-algebra structure after a finite amount of computational work. A-infinity, strong homotopy associativity, inductive computation 17A42; 17-04 1.