Let X be an irreducible affine algebraic curve over the complex numbers, and let D(X) denote its ring of differential operators. This project is concerned with determining for two irreducible affine curves X and Y, when D(X) and D(Y) being isomorphic implies that X and Y are isomorphic. The approach to this problem will involve finding the maximal commutative ad-nilpotent subalgebras of D(X) and then determining if the coordinate ring of Y being isomorphic to an ad-nilpotent subalgebra of D(X) implies that D(X) and D(Y) are isomorphic. This project is in the general area of ring theory. A ring is an algebraic object with an addition and multiplication defined on it. This particular project is concerned with a special ring call the ring of differential operators of an affine curve. These rings are important in several areas of mathematics and physics.
