These notes are based on a series of lectures given by the first author at the school of 'Poisson 2010', held at IMPA, Rio de Janeiro. They contain an exposition of the theory of super-and graded manifolds, cohomological vector fields, graded symplectic structures, reduction and the AKSZ-formalism.

Communicated by I. Moerdijk MSC: 16E45 17B63 17B70 17B81 18G55 53D17 58C50 a b s t r a c t This note elaborates on Th. Voronov's construction [Th. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (1-3) (2005) 133-153; Th. Voronov, Higher derived brackets for arbitrary derivations, Travaux Math. XVI (2005) 163-186] of L ∞ -structures via higher derived brackets with a Maurer-Cartan element. It is shown that gauge equivalent Maurer-Cartan elements induce L ∞ -isomorphic structures. Applications in symplectic, Poisson and Dirac geometry are discussed.

We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an functor from the representations up to homotopy of a Lie algebroid A to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several examples including Lie algebras and Poisson manifolds. The construction is based on an version of de Rham's theorem due to Gugenheim [15]. The integration procedure we explain here amounts to extending the construction of parallel transport for superconnections, introduced by Igusa [17] and Block-Smith [6], to the case of certain differential graded manifolds

We present a connection between the BFV-complex (appreviation for Batalin-Fradkin-Vilkovisky complex) and the so-called strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are L∞ quasi-isomorphic and control the same formal deformation problem.However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic submanifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instead.

We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets.