Commuting Differential Operators with Elliptic Coefficients

Ordinary Differential Operators with elliptic coefficients have strong geometric properties and a long history which received new impetus in the context of integrable hierarchies. Partial Differential Operators with the same property, on the contrary, have hardly been investigated. We bring together results on the former, together with many open problems, and an extension to hyperelliptic coefficients; and propose and exemplify new methods on the latter. In both contexts, our emphasis is on maximal commuting algebras of operators over the complex numbers.