We extend Milnor's mu-invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for mu-invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves corresponds to axioms of Loday's diassociative algebra. The relation of tangles to diassociative algebras is formulated in terms of a morphism of corresponding operads.

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer-Cartan equations on corresponding governing differential graded Lie algebras. Cohomology theories of all these structures are described in a concise way using the big bracket construction of Kosmann-Schwarzbach. This approach provides a definition of an L_\infty-(quasi)bialgebra (strong homotopy Lie (quasi)bialgebra). We recover an L_\infty-algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L_\infty (quasi)bialgebra structure on V as a differential operator Q on V, self-commuting with respect to the Poisson bracket. Finally, we establish an L_\infty-version of a Manin (quasi) triple and get a correspondence theorem with L_\infty-(quasi) bialgebra.

We show that a commutative algebra $A$ with any square zero odd differential operator is a natural generalization of the Batalin-Vilkovisky algebra. While such an operator of order $2$ defines the Lie structure on $A$, an operator of an order higher than $2$ (Koszul-Akman definition ) leads to the structure of a Lie up-to homotopy algebra on $A$. This allows us to propose a definition of a Batalin-Vilkovisky algebra up-to homotopy. We also make an important conjecture generalizing Kontsevich formality theorem to the BV-algebra level.

This is an expository note on Fedosov's construction of deformation quantization. Given a symplectic manifold and a connection on it, we show how to   calculate the star-product step by step. We draw simple diagrams to solve the recursive equations for the Fedosov connection and for flat sections of the Weyl algebra bundle corresponding to  functions.  We also reflect on the differences of symplectic and Riemannian geometries.

A symplectic fibration is a fibre bundle in the symplectic category. We find the relation between deformation quantization of the base and the fibre, and the total space. We use the weak coupling form of Guillemin, Lerman, Sternberg and find the characteristic class of deformation of symplectic fibration. We also prove that the classical moment map could be quantized if there exists an equivariant connection. Along the way we touch upon the general question of quantization with values in a bundle of algebras.

We consider a Hamiltonian action of a group on a symplectic manifold with a given moment map. Provided with an equivariant connection on this manifold we show that the action in the deformed algebra is the same, namely by the Poisson bracket. We employ Fedosov's construction of deformation quantization.