RUI: Noncommutative polynomial algebras and the foundations of noncommutative geometry

数学中一个反复出现的主题是，许多类型的几何空间可以通过定义在该空间上的函数的集合来精确地刻画。这些函数可以相加和相乘，形成一种称为代数的结构。这些函数代数的定义方式是，两个函数的乘积不依赖于它们相乘的顺序，就像在数值乘法中一样。但在数学和量子物理中出现的许多代数是非对易的，并且这些代数没有对应的几何空间(在我们目前对这个术语的理解中)。这个项目有两个相关的目标：第一，更好地理解一类代数，它应该对应于普通“平”(仿射)空间的量子版本；第二，寻找一个合适的类似空间的物体，它应该形成非对易代数的几何对应。提出的方法将促进环理论、泛函分析和量子物理之间的相互作用。PI将包括本科生参与这项研究。这个项目解决了非对易代数的两个目标，这两个目标都与非对易几何的基础问题有关。第一个目标是发展研究斜(或“扭曲”)Calabi-Yau代数的新方法。这些代数同时推广了非交换代数几何中的两类重要代数：Calabi-Yau代数和Artin-Schelter正则代数。要考虑的问题包括这些代数上作用的不变量，这些代数在低维上的结构，以及这些代数上的模的对偶理论。第二个目标是为非交换代数构造一个有意义地扩展交换环和代数的谱的函数式不变量。这将建立在对几个自然的非对易谱函子候选者的严格障碍之上。最初的重点将是将Gelfand谱函子从交换C*-代数推广到非交换C*-代数。

A recurring theme in mathematics is that many kinds of geometric spaces can be precisely characterized by the collection of functions defined on the space. These functions can be added and multiplied, forming a structure called an algebra. These algebras of functions are defined in such a way that the product of two functions does not depend upon the order in which they are multiplied, just as in numerical multiplication. But many algebras arising in mathematics and quantum physics are noncommutative, and such algebras do not have a corresponding geometric space (in our current understanding of this term). This project has two related goals: first, to better understand a certain class of algebras that should correspond to quantum versions of ordinary "flat" (affine) spaces, and second, to seek a suitable space-like object that should form the geometric counterpart of a noncommutative algebra. The proposed approach will promote interaction between ring theory, functional analysis, and quantum physics. The PI will involve undergraduate students in this research.This project addresses two objectives in noncommutative algebra that are both related to foundational issues in noncommutative geometry. The first objective is to develop new techniques for the study of skew (or "twisted") Calabi-Yau algebras. These algebras simultaneously generalize two important classes of algebras from noncommutative algebraic geometry: Calabi-Yau algebras and Artin-Schelter regular algebras. Problems to be considered include invariants of actions on these algebras, the structure of these algebras in low dimensions, and duality theory for modules over such algebras. The second objective is to construct a functorial invariant for noncommutative algebras that meaningfully extends the spectrum of commutative rings and algebras. This will build upon rigorous obstructions to several natural candidates for noncommutative spectrum functors. The initial focus will be to extend the Gelfand spectrum functor from commutative to noncommutative C*-algebras.