Geometric Insights in Noncommutative Algebra

交换代数研究的是与乘法顺序无关的代数系统，而非交换代数允许两个乘积XY和YX可能不相同。非交换代数历史上出现在对称的代数研究中，在线性代数中，也许在量子力学中最深刻。自从代数几何出现以来，交换代数就被几何思想彻底改变了。同样，在过去的几十年里，非交换几何的发展激发了非交换代数的许多新思想和新观点，但迄今为止，它提出的挑战比它解决的要多。例如，许多非交换代数由于其构造的几何性质而被期望表现良好，但我们缺乏代数工具来证明这一点。此外，非交换几何在很大程度上仍然是松散相关框架的集合，这些框架彼此之间不直接兼容，这使得该领域对新手来说特别困难。该项目将通过创造新的技术来推导几何构造的非交换代数的良好代数性质，并提供新的工具来表示与非交换代数对应的几何结构，从而直接解决这些基本问题。该项目将为研究生提供研究和培训机会。本计画的第一部分主要讨论非交换代数几何中的同调问题。PI和合作者将使用Koszul对偶方法和有限维代数的表示来攻击一个长期存在的猜想，即Artin-Schelter （AS）正则代数是域。相关的技术将被用来理解什么时候广义AS正则代数不一定是连通的，如分级Calabi-Yau代数，是素环。本计画的第二部分聚焦于非交换几何中的谱问题。PI将使用各种方法来理解非交换离散空间，并利用它们来构建环和C*-代数的非交换谱函子。研究的主题包括非交换空间的结构束、对偶代数的表征方法、C*-代数的离散化以及环的射影表示理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Commutative algebra studies algebraic systems for which the order of multiplication is irrelevant, while noncommutative algebra allows for the possibility that the two products XY and YX may not be the same. Noncommutative algebra historically arose in the algebraic study of symmetry, in linear algebra, and perhaps most profoundly in quantum mechanics. Since the advent of algebraic geometry, commutative algebra has been revolutionized by geometric ideas. In a similar way, the development of noncommutative geometry over the past several decades has stimulated many new ideas and perspectives in noncommutative algebra, but to date it has posed more challenges than it has solved. For instance, many noncommutative algebras are expected to be well-behaved due to the geometric nature of their construction, but we lack the algebraic tools to prove that this is the case. Furthermore, noncommutative geometry largely remains a collection of loosely related frameworks that are not directly compatible with one another, which can make the field especially difficult for newcomers. This project will directly address these fundamental problems by creating new techniques to deduce good algebraic properties for geometrically constructed noncommutative algebras and providing new tools to represent geometric structures corresponding to noncommutative algebras. The project will involve research and training opportunities for graduate students.The first part of this project focuses on homological problems in noncommutative algebraic geometry. The PI and collaborators will use methods of Koszul duality and the representations of finite-dimensional algebras to attack a longstanding conjecture that Artin-Schelter (AS) regular algebras are domains. Related techniques will be used to understand when generalized AS regular algebras that are not necessarily connected, such as graded Calabi-Yau algebras, are prime rings. The second part of this project is focused on spectral problems in noncommutative geometry. The PI will use a variety of approaches to understand noncommutative discrete spaces and utilize them to construct noncommutative spectrum functors for both rings and C*-algebras. Topics to be investigated include structure sheaves for noncommutative spaces, methods to characterize dual coalgebras, discretization of C*-algebras, and a projective representation theory for rings.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.