Noncommutative Geometry and Rings of Differential Operators

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Principal Investigator: J. Tobias Stafford Proposal Number: 0245320Institution: University of Michigan Ann ArborNONCOMMUTATIVE GEOMETRY AND RINGS OF DIFFERENTIAL OPERATORS PROJECT ABSTRACT: There are two main themes to Professor Stafford's research project. The first concerns the theory of "noncommutative projective geometry", or the application of projective algebraic geometry to the study of noncommutative graded rings. The underlying project is to appropriately classify noncommutative surfaces (equivalently, noncommutative graded algebras of Gelfand-Kirillov dimension 3), thereby extending the classification, by Professors Artin and Stafford, of noncommutative curves (graded algebras of Gelfand-Kirillov dimension 2). In particular Professor Stafford intends to study noncommutative analogues of blowing up and down. Jointly with Professors Keeler and Rogalski, Professor Stafford has used projective algebraic geometry to create noetherian connected graded algebras that are not strongly noetherian. The algebraic geometrical perspective may lead to further insights regarding such algebras. Professor Stafford will also work on classifying moduli spaces of vector bundles and modules over various noncommutative algebras. The second main theme, the study of rings of differential operators and their invariants, includes applications to the representation theory of Lie algebras. The techniques developed in this project should be useful for understanding the rational Cherednik algebras and their spherical subalgebras, as introduced by Professors Etingof and Ginzburg.Much of mathematics is inherently noncommutative; perhaps the most famous example is Heisenberg's uncertainty principle: Measuring the position and then measuring the momentum of a particle gives a different answer than first measuring the momentum of the particle and then the position. This noncommutativity naturally leads to the study of noncommutative algebras. Much of Professor Stafford's research and of the present project is in the area of noncommutative geometry, a theory that uses the techniques and intuition of algebraic geometry to study such algebras. This theory has led to a classification of noncommutative analogues of curves, and the motivating question behind the present project is to extend those results to provide a deeper understanding of noncommutative analogues of surfaces.