Correspondences on curves, Fermat quotients, and uniformization

The quotient of a curve by a correspondence usually reduces to a point in algebraic geometry. One way to fix this pathology is to extend algebraic geometry in a "non-commutative" direction. Another way (which is the subject of this talk) is to extend algebraic geometry by staying within the commutative setting but adjoining instead a new operation: the Fermat quotient. It turns out that in this new geometry a number of interesting quotients of curves by correspondences become non-trivial and indeed rather interesting. The examples that can be treated in this way arise from correspondences that can be "analytically uniformized". Remarkably these are closely related to some of the main examples treated via non-commutative geometry.