Noncommutative Geometry and Rings of Differential Operators

主要研究人员：J.Tobias Stafford建议编号：0245320机构：密歇根大学Ann ArborNONCommuTATIVE微分算子几何和环项目摘要：斯塔福德教授的研究项目有两个主要主题。第一种是“非对易射影几何”理论，即射影代数几何在非对易分次环研究中的应用。基本的项目是适当地分类非交换曲面(等价地，Gelfand-Kirillov维为3的非交换分次代数)，从而扩展Artin和Stafford教授对非交换曲线(Gelfand-Kirillov维为2的分次代数)的分类。具体地说，斯塔福德教授打算研究吹气和吹气的非对易类比。斯塔福德教授与Keeler和Rogalski教授合作，使用射影代数几何创建了Notherian连通分次代数，这些代数不是强Notherian的。代数几何视角可能会导致对此类代数的进一步洞察。Stafford教授还将致力于对各种非交换代数上的向量丛和模的模空间进行分类。第二个主要主题是研究微分算子环及其不变量，包括在李代数表示理论中的应用。正如Etingof和Ginzburg教授所介绍的，在这个项目中发展的技巧应该有助于理解有理Cherednik代数及其球子代数。许多数学本质上是非对易的；也许最著名的例子是海森伯格的测不准原理：测量一个粒子的位置，然后测量粒子的动量，给出一个不同的答案，而不是先测量粒子的动量，然后测量位置。这种非交换性自然地导致了对非交换代数的研究。斯塔福德教授和本项目的大部分研究都是在非对易几何领域，这是一种利用代数几何的技巧和直觉来研究此类代数的理论。这一理论导致了曲线的非对易相似的分类，而本项目背后的激励问题是扩展这些结果以提供对曲面的非对易相似的更深层次的理解。

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.