Non-commutative schemes associated to semi-graded rings

In this talk we will introduce the semi-graded rings and we will discuss the Serre-Artin-Zhang-Verevkin theorem about non-commutative schemes associated to semi-graded rings. This new class of non-commutative rings generalizes finitely graded algebras and skew PBW (Poincaré-Birkhoff-Witt) extensions.