Mathematical Sciences: Rings of Differential Operators and Enveloping Algebras

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This research will focus on the study of differential operators on algebraic varieties and the connections with primitive ideals in the enveloping algebra of a semi-simple Lie algebra. Given a curve, there is a finite dimensional algebra (a certain factor ring of the ring of differential operators) associated to each singular point of the curve. A general problem is to understand the precise relation between the local geometry at a singular point, and the structure of the ring of differential operators. This research is in the general area of noncommutative ring theory. The rings considered in this project are of great interest in many parts of mathematics. Given a curve, one of these rings can be associated with certain points on the curve. A better understanding of these rings and this association will be useful in determing the geometry of a given curve.