题目：Commuting partial differential operators and higher-dimensional algebraic varieties in connection with higher-dimensional analogues of the KP theory
摘要：The well-known KP hierarchy is an infinite system of nonlinear partial differential equations, which describes, among other things, the isospectral deformations of the rings of commuting ordinary differential operators. In geometric terms, these deformations are described as flows on the Jacobians of the spectral curves of such rings, which can also be regarded as restriction of the flows defined by the hierarchy on the Sato Grassmannian. I will talk about the analogue of this theory in the two-dimensional case. In this case, rings of commuting differential operators of two variables, or more general rings of differential-difference or pseudodifferential operators, and their isospectral deformations are considered. Deformations are described by analogues of the KP hierarchy — the modified Parshin hierarchies, which define flows on the moduli space of torsion-free sheaves of a spectral surface.