Basic introduction to few aspects of Derived Algebraic Geometry

This talk is not a detailed and precise exposition on DAG, but it is conceived more as a kind of advertisement on this theory and some of its interesting new features one should contemplate and try to understand, as they might reveal interesting new insights also on classical objects. We select some of the several motivations for introducing it (non-representability of moduli problem and non-naturality of the obstruction theory), and then we will go through the homotopy theory of simplicial commutative algebras and their cotangent complex. We will introduce the category of derived schemes and we will describe their relation with classical schemes. A good amount of time will be dedicated to examples.