We develop a formalism involving Atiyah classes of sheaves on a smooth manifold, Hochschild chain and cochain complexes. As an application we prove a version of the Riemann-Roch theorem.

Let G be a semisimple complex Lie group with a Borel subgfoup B. Let X = G/B be the flag manifold of GThe moduli space of G-monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. It was recently explicitly computed by R. Bielawski in case G = SL n . We propose a simple explicit formula for another natural symplectic structure on M b (X, α). It is tantalizingly similar to R. Bielawski's formula, but in general (rank > 1) the two structures do not coincide. Let P ⊃ B be a parabolic subgroup. The construction of the Poisson structure on M b (X, α) generalizes verbatim to the space of based maps M = M b (G/P, β). In most cases the corresponding map T * M → T M is not an isomorphism, i.e. M splits into nontrivial symplectic leaves. These leaves are explicitly described.

We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/∆ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator.These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.

Abstract. We introduce Weyl n-algebras and show how their factorization complex may be used to define invariants of manifolds. In the appendix, we heuristically explain why these invariants must be perturbative Chern-Simons invariants.

Abstract. Given a Lie algebra with a scalar product, one may consider the latter as a symplectic structure on a dg-scheme, which is the spectrum of the Chevalley-Eilenberg algebra. In the first section we explicitly calculate the first order deformation of the differential on the Hochschild complex of the Chevalley-Eilenberg algebra. The answer contains the Duflo character. This calculation is used in the last section. There we sketch the definition of the Wilson loop invariant of knots, which is, hopefully, equal to the Kontsevich integral, and show that for unknot they coincide. As a byproduct, we get a new proof of the Duflo isomorphism for a Lie algebra with a scalar product.