Masterclass: Elliptic Algebras and Poisson Structures

I will survey the theory of associative $\mathbb{Z}\geq 0$ -graded algebras presented by $n$ generators and $n(n-1)/2$ quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). We will consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve and a point on this curve) which are flat deformations of the polynomial ring in $n$ variables. We will discuss geometry of the corresponding Poisson structures. Diverse properties of these algebras and Poisson structures will be described, together with their relations to integrable systems, deformation quantization, moduli spaces and other directions of modern investigations.