The Schur–Sato theory for quasi-elliptic rings and some of its applications

The notion of quasielliptic rings appeared as a result of an attempt to classify a wide class of commutative rings of operators found in the theory of integrable systems, such as rings of commuting differential, difference, differential-difference, etc. operators. They are contained in a certain non-commutative “universe” ring — a purely algebraic analogue of the ring of pseudodifferential operators on a manifold, and admit (under certain mild restrictions) a convenient algebraic-geometric description. An important algebraic part of this description is the Schur–Sato theory — a generalisation of the well known theory for ordinary differential operators. I'll talk about this theory in dimension n and about some of its unexpected applications related to the generalized Birkhoff decomposition and to the Abhyankar formula.