Noncommutative Geometry and Rings of Differential Operators

非交换几何和微分算子环

• 批准号: 0245320
• 负责人: J. Tobias Stafford
• 金额: $ 23万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245320&HistoricalAwards=false
• 关键词: Noncommutative; Geometry; Rings; Differential; Operators

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.

摘要：Stafford教授的研究项目主要有两个主题。第一部分是关于“非交换射影几何”的理论，或者说是射影代数几何在非交换梯度环研究中的应用。基础项目是对非交换曲面（相当于Gelfand-Kirillov维数为3的非交换渐变代数）进行适当分类，从而扩展了Artin和Stafford教授对非交换曲线（Gelfand-Kirillov维数为2的渐变代数）的分类。斯塔福德教授特别打算研究爆炸和爆炸的非交换类似物。与Keeler教授和Rogalski教授合作，Stafford教授使用投影代数几何创建了非强诺etherian的noether连通梯度代数。代数几何透视可能会导致对这些代数的进一步见解。Stafford教授还将研究向量束的模空间和各种非交换代数上的模空间的分类。第二个主题，微分算子环及其不变量的研究，包括在李代数表示理论中的应用。在这个项目中开发的技术应该有助于理解有理Cherednik代数及其球面子代数，正如Etingof和Ginzburg教授所介绍的那样。许多数学本质上是不可交换的；也许最著名的例子是海森堡的测不准原理：先测量粒子的位置，然后测量粒子的动量，得到的答案与先测量粒子的动量，然后再测量粒子的位置得到的答案不同。这种非交换性自然导致了对非交换代数的研究。斯塔福德教授的大部分研究和目前的项目都是在非交换几何领域，这是一种利用代数几何的技术和直觉来研究代数的理论。这一理论导致了曲线的非交换类似物的分类，而本项目背后的激励问题是扩展这些结果，以提供对曲面的非交换类似物的更深层次的理解。