Spectral Invariants of Noncommutative Spaces

非交换空间的谱不变量

• 批准号: RGPIN-2019-04748
• 负责人: Khalkhali, Masoud
• 金额: $ 1.53万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758530
• 关键词: Spectral; Invariants; Noncommutative; Spaces

My proposed research aims to extend the very notion of space and its geometry, and study the curvature (bending) of these new types of spaces. As such it can have potential implications for our current understanding of the fabric of spacetime in physics, and for mathematics itself. In fact it is well known that the current notion of spacetime as a smooth 4-dimensional manifold in Einstein's general theory of relativity is not adequate for quantum gravity and has to be replaced, in very small distances and high energies, by a quantum spacetime. It is just not clear what the replacement exactly will be! There are several proposals, but I am convinced our's offers the best hope. The internal needs of mathematics, on the other hand, has also suggested these new spaces and their geometry, called noncommutative geometry (NCG), simply  because we need to treat highly singular spaces of leaves of a foliation, a fractal, or bad quotients of nice spaces, by tools of geometry, analysis, and topology. History of mathematics is the history of the evolution of the notion of space too! Starting with Euclid's elements, the work of Descartes, Gauss, Riemann, Hilbert, von Neumann, Gelfand, Grothendieck, Connes and Kontsevich, has systematically extended the notion of space and its geometry to ever more complex paradigms and now to NC dimensions. This proposal is one of the latest, and as far as the study of curvature  goes, the latest, chapter in this saga. Mathematically, I am building on, and extending, the work of Hermann Weyl and its vast extension by Kac, Milnor, Gilkey, McKean-Singer, Atiyah-Bott, and Connes, summarized under the title of spectral geometry, where one shows that one can hear (many things about) the shape of a drum! Here the words shape and drum must be broadly understood! Thus a drum can be a domain in a Euclidean space, a compact manifold, a fractal, or, as in this proposal, a noncommutative manifold. Similarly, shape could be understood as volume, scalar curvature,  Ricci curvature, and, in general any concept  that can be captured in terms of the  spectrum of operators like Laplace or Dirac and their more exotic noncommutative analogues a la Connes. What we have been able to prove in the past 10 years is that one can in fact hear a lot of geometry of  NC spaces too and in fact  spectral geometry ideas is the only way that one can gain  information about the intrinsic geometry of these new curved NC spaces. Beyond spectral geometry, I am  now beginning to incorporate  very new exciting ideas suggested by random matrix theory and topological recursion into the study of NC spaces.  I will extend my study of scalar and Ricci curvature of NC spaces to higher dimensions and to nonconformal metrics. One goal of this proposal is to eventually define something like the full Riemann curvature in NCG and to prove a Gauss-Bonnet type theorem in all even dimensions. All signs indicate that this is within the reach of  our methods.

我提出的研究旨在扩展空间及其几何的概念，并研究这些新型空间的曲率（弯曲），因此它可能对我们目前对物理学中时空结构的理解产生潜在的影响。事实上，众所周知，当前爱因斯坦广义相对论中时空作为平滑的 4 维流形的概念对于量子引力来说是不够的，在非常短的距离和高能量下，必须被量子引力所取代。量子只是不清楚替代品到底是什么！有几个建议，但我相信我们的建议提供了最好的希望，另一方面，也提出了这些新空间及其几何形状，称为非交换几何（NCG），仅仅是因为我们需要通过几何、分析和拓扑工具来处理叶子的高度奇异空间、分形或良好空间的坏商。的演变空间的概念也从欧几里得的元素开始，笛卡尔、高斯、黎曼、希尔伯特、冯·诺依曼、格尔凡德、格洛腾迪克、康尼斯和康采维奇的工作，系统地将空间及其几何的概念扩展到更加复杂的范式，现在这个提议是最新的，就曲率研究而言，是我正在构建的这个传奇的最新章节。赫尔曼·韦尔（Hermann Weyl）的工作及其对 Kac、Milnor、Gilkey、McKean-Singer、Atiyah-Bott 和 Connes 的巨大扩展的研究和扩展，总结在光谱几何的标题下，其中人们表明人们可以听到（许多东西）关于）鼓的形状！这里必须广泛地理解“形状”和“鼓”这两个词！因此，鼓可以是欧几里得空间中的域、紧流形、分形，或者如本提案中那样，非交换流形。形状可以理解为体积、标量曲率、里奇曲率，以及一般来说可以用拉普拉斯或狄拉克等算子的谱来捕捉的任何概念，以及它们更奇特的非交换类似物，如康涅斯。我们在过去 10 年里已经能够证明的是，人们实际上也可以听到很多 NC 空间的几何知识，而且事实上谱几何思想是唯一的除了谱几何之外，我现在开始将随机矩阵理论和拓扑递归提出的非常新的令人兴奋的想法融入到 NC 空间的研究中。研究 NC 空间的标量曲率和 Ricci 曲率到更高维度和非共形度量该提案的一个目标是最终定义类似于 NCG 中的完整黎曼曲率的东西并证明所有偶维中的高斯-博内特型定理都表明这在我们的方法的范围内。