Mathematical Sciences: Rings of Differential Operators

设X是复数上的一条不可约仿射代数曲线，D(X)表示它的微分算子环。本文研究了当D(X)和D(Y)同构时，确定两条不可约仿射曲线X和Y时，X和Y是同构的。解决这个问题的方法包括找到D(X)的最大可交换幂零子代数，然后确定Y的坐标环与D(X)的幂零子代数同构是否意味着D(X)和D(Y)同构。这个项目属于环理论的一般领域。环是一个代数对象，在其上定义了加法和乘法。这个特殊的项目涉及一个特殊的环，称为仿射曲线的微分算子环。这些环在数学和物理的几个领域都很重要。

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.