In this talk, we discuss the reduction-quantization diagram in terms of formality. First, we propose a reduction scheme for multivector fields and multidifferential operators, phrased in terms of L-infinity morphisms. This requires the introduction of equivariant multivector fields and equivariant multidifferential operator complexes, which encode the information of the Hamiltonian action, i.e., a G-invariant Poisson structure allowing for a momentum map. As a second step, we discuss an equivariant version of the formality theorem, conjecturedby Tsygan and recently solved in a joint work with Nest, Schnitzer, and Tsygan. This result has immediate consequences in deformation quantization, since it allows for obtaining a quantum moment map from a classical momentum map with respect to a G-invariant Poisson structure.[-]

We prove the statement$/$conjecture of M. Kontsevich on the existence of the logarithmic formality morphism $\mathcal{U}^{log}$. This question was open since 1999, and the main obstacle was the presence of $dr/r$ type singularities near the boundary $r = 0$ in the integrals over compactified configuration spaces. The novelty of our approach is the use of local torus actions on configuration spaces of points in the upper half-plane. It gives rise to a version of Stokes' formula for differential forms with singularities at the boundary which implies the formality property of $\mathcal{U}^{log}$. We also show that the logarithmic formality morphism admits a globalization from $\mathbb{R}^{d}$ to an arbitrary smooth manifold.[-]

Let $\mathfrak{h}$ be a finite dimensional real Leibniz algebra. Exactly as the linear dual space of a Lie algebra is a Poisson manifold with respect to the Kostant-Kirillov-Souriau (KKS) bracket, $\mathfrak{h}^*$ can be viewed as a generalized Poisson manifold. The corresponding bracket is roughly speaking the evaluation of the KKS bracket at $0$ in one variable. This (perhaps strange looking) bracket comes up naturally when quantizing $\mathfrak{h}^*$ in an analoguous way as one quantizes the dual of a Lie algebra. Namely, the product $X \vartriangleleft Y = exp(ad_X)(Y)$ can be lifted to cotangent level and gives than a symplectic micromorphism which can be quantized by Fourier integral operators. This is joint work with Benoit Dherin (2013). More recently, we developed with Charles Alexandre, Martin Bordemann and Salim Rivire a purely algebraic framework which gives the same star-product.[-]