Dr. Natalja Iyudu: Sklyanin algebras via Groebner bases and finiteness conditions for potential algebras

Sklyanin algebras were introduced in 1983 in the work related to Yang-Baxter equations and representations of quantum algebras. They were extensively studied thereafter from different points of view: in integrable systems, algebraically, etc. We shall recall some well-known algebraic results about the Sklyanin algebras (Artin, Schelter, Van den Bergh), introducing new combinatorial techniques related to the Groebner bases theory. Namely, we calculate the Hilbert series, prove Koszulity, Poincare-Birkhoff-Witt property, Calabi-Yau property  etc., depending on parameters of the Sklyanin algebras. Similar methods are then used for generalized Sklyanin algebras and other potential algebras, which are important invariants in the non-commutative resolution of singularities.