Schur-Sato Theory for Quasi-Elliptic Rings and Some of Its Applications
Time: 2022-10-21
Published By: Wenqiong Li
Speaker(s): Alexander Zheglov (Lomonosov Moscow State University)
Time: 14:00-15:00 October 25, 2022
Venue: Room 29, Quan Zhai, BICMR
The Schur-Sato theory, which will be discussed in the talk, is a generalization of a well-known theory in dimension one for rings of ordinary differential operators. We develop it for a wide class of so-called quasi-elliptic rings in arbitrary dimension. Such rings have been defined in order to classify a wide class of commutative rings of operators encountered in the theory of integrable systems (such as, for example, rings of commuting differential, differential, differential-difference and etc. Operators). The theory is applied to classification of quasi-elliptic rings in terms of some subspaces (Schur pairs). This theory is also one of steps toward the higher dimensional Krichever correspondence — a higher-dimensional analogue of the well-known and fruitful interrelation between KdV- or, more general, KP-equations, and algebraic curves with additional geometric data on the other side. In addition, it has several other applications: among them a new proof of the Abyankar formula and a new proof of the generalized Birkhoff theorem.
