I will present the main objective of the series of lectures, namely the construction of deformation quantization of certain moduli spaces. In a second part I will present some of the basic notions of derived algebraic geometry, such as derived schemes and derived algebraic stacks.

Branes are extended objects that define the boundary conditions of sigma models. These lectures cover the geometry of branes, their roles in duality and mirror symmetry, and the relation to quantization.

Branes are extended objects that define the boundary conditions of sigma models. These lectures cover the geometry of branes, their roles in duality and mirror symmetry, and the relation to quantization.

In this second lecture I will present more about derived algebraic geometry and will introduce the notion of shifted symplectic structures. I will state several existence theorems and deduce that many moduli spaces, when suitably considered as derived algebraic stacks, are endowed with natural shifted symplectic structures.

This last lecture is concerned with the construction of deformation quantization of moduli spaces endowed with shifted symplectic structures. More generally, I will present the notion of shifted Poisson structures as well as a shifted version of Kontsevich's formality theorem. I will explain how this implies the existence of quantizations. The lecture will end with examples, some recovering well known quantum objects (e.g. quantum groups), and some new.

Branes are extended objects that define the boundary conditions of sigma models. These lectures cover the geometry of branes, their roles in duality and mirror symmetry, and the relation to quantization.