In this paper, defining Poisson functions on super manifolds, we show that the graphs of Poisson functions are Dirac structures, and find Poisson functions which include as special cases both quasi-Poisson structures and twisted Poisson structures.

Suppose that the compact and connected Lie group $G$ acts holomorphically on the irreducible complex projective manifold $M$, and that the action linearizes to the Hermitian ample line bundle $L$ on $M$. Assume that $0$ is a regular value of the associated moment map. The spaces of global holomorphic sections of powers of $L$ may be decomposed over the finite dimensional irreducible representations of $G$. We study how the holomorphic sections in each equivariant piece asymptotically concentrate along the zero locus of the moment map. In the special case where $G$ acts freely on the zero locus of the moment map, this relates the scaling limits of the Szego kernel of the quotient to the scaling limits of the invariant part of the Szego kernel of $(M,L)$.

Many interesting $C∗$-algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution $C∗$-algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, including geometric quantization of symplectic manifolds and the $C∗$-algebra of a Lie groupoid. I sketch a few new examples, including twisted groupoid $C∗$-algebras as quantizations of bundle affine Poisson structures.