Noncommutative Algebra

本文讨论了非交换代数领域中的两个主题，即非交换射影几何和非交换Iwasawa代数理论。首席研究员建议从事该领域的一些核心项目，如Artin关于非交换射影曲面的计划和量子射影三维空间的分类（或全局四维的连通分级Artin- schelter正则代数的分类）。主要研究非交换Iwasawa代数的模理论，了解非交换Iwasawa代数的代数面与数论面之间的联系。该建议的另一个目标是寻找反映和预测非交换代数结构的各种环论和同调不变量。不可交换性是数学和其他学科中一个非常普遍且日益重要的现象。非交换代数是表征和理解非交换性的数学基础。随着非交换性的更多细节的揭示，研究非交换代数的结构是必要的。非交换代数几何的最新进展为非交换代数的研究提供了新的思路和方法。该提案的一部分强调了使用代数几何技术的非交换代数的分类问题。该项目也适用于其他学科，如代数几何、数论和数学物理，并将增进对这些学科的更广泛理解。

The proposal concerns two topics in the area of noncommutative algebra, namely, noncommutative projective geometry and the theory of noncommutative Iwasawa algebras. The principal investigator proposes to work on some central projects in the field such as Artin's program on noncommutative projective surfaces and the classification of quantum projective three-spaces (or the classification of connected graded Artin-Schelter regular algebras of global dimension four). The principal investigator proposes to study the module theory of noncommutative Iwasawa algebras and to understand the connection between the algebraic side and the number-theoretic side of the noncommutative Iwasawa theory. Another goal of the proposal is to search for various ring-theoretic and homological invariants that reflect and predict the structure of noncommutative algebras.Noncommutativity is a very common and increasingly important phenomenon in Mathematics and other disciplines. The subject of noncommutative algebra is the mathematical foundation for characterizing and understanding noncommutativity. As more details of noncommutativity are to be revealed, it is essential to examine the structure of noncommutative algebras. Recent developments in noncommutative algebraic geometry furnish new ideas and methods for the study of noncommutative algebras. Part of the proposal emphasizes some classification problems of noncommutative algebras using techniques from algebraic geometry. The project has applications to other subjects such as algebraic geometry, number theory and mathematical physics, and will enhance broader understanding of these subjects.