The Anick conjecture on the minimal Hilbert series of quadratic algebras

We present several results on the Anick conjecture which asserts that the lower bound for the Hilbert series, known as the Golod-Shafarevich estimate is attained on generic quadratic algebra. The technique (due to Anick), allowing to write down precisely the formula for the Hilbert series will be demonstrated. We will discuss also related questions of Koszulity and being noncommutative complete intersection (NCCI). Connections to the latter property on the level of finite dimensional representations, namely, introduced by Ginsburg and Etingof notion of representational complete intersection (RCI) will be considered and some examples given.