Graded Algebras, Polynomial Identities and Generic Constructions

In these lecture we present some results which intertwine topics as graded algebras, polynomial identities and algebras of generic elements. Some of these connection are classical and well known to different communities (e.g. crossed produts, Galois cohomology, algebra of generic matrices, general group gradings on finite dimensional algebra). Some other connection among these topics are relatively new where these are realized via the theory of group graded polynomial identities. In particular, using (G-graded) asymptotic PI theory, we outline the proof of a conjecture of Bahturin and Regev on regular gradings on associative algebras. These lectures took place in Porto Cesareo, Italy. The author is mostly grateful to the organizers of the meeting Advances in Group theory.