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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737833
• 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大空间及其几何形状的概念，并研究这些新型空间的咖喱（弯曲）。因此，它可能对我们目前对物理时空结构以及数学本身的理解具有潜在的影响。实际上，众所周知，当前时空的概念是爱因斯坦相对论的一般一般理论中平滑的四维流形的概念不足以适合量子重力，并且必须以很小的距离和高能量替换为量子时空。只是不清楚替换的准确是什么！有几个建议，但我坚信我们的最大希望。另一方面，数学的内部需求还提出了这些新空间及其几何形状，称为非交通性几何形状（NCG），仅仅是因为我们需要通过工具，分析和拓扑来处理叶面的高度奇异的叶子，分形或不良空间的良好引号，不错的空间。数学历史也是空间概念发展的历史！从欧几里得的元素开始，笛卡尔，高斯，里曼，希尔伯特，冯·诺伊曼，盖凡德，盖凡德，格罗瑟迪克，康纳斯和肯特塞维奇的作品已系统地将太空及其几何形状的概念及其几何形状扩展到了更加复杂的范式，现在可以扩展到NC尺寸。该提议是最新的，就曲率研究而言，该传奇中的最新一章。从数学上讲，我正在建设并扩展了Hermann Weyl及其巨大延伸的作品，由KAC，Milnor，Gilkey，McKean-Singer，Atiyah-Bott和Connes和Connes汇总为光谱几何学的标题，其中一个人可以在其中显示（很多事情）（许多事情）关于鼓的形状！在这里，必须广泛理解形状和鼓的单词！鼓可以是欧几里得空间中的一个域，紧凑的歧管，分形，或者像在此提案中一样，是一个非交换歧管。同样，形状可以被视为体积，标态曲率，RICCI曲率，并且通常可以根据LaPlace或Dirac等操作员及其更奇异的非交通性类似物A La Connes捕获的任何概念。在过去的十年中，我们能够证明的是，实际上，人们也可以听到许多NC空间的几何形状，实际上，光谱几何思想是唯一可以获取有关这些新弯曲NC空间内在几何形状的信息的唯一方法。除了光谱几何形状之外，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纳入NC空间的研究中。我将将NC空间标量和RICCI曲率的研究扩展到更高的维度和非符号指标。该提案的目标之一是最终定义诸如NCG中完整的Riemann Curvature之类的内容，并在所有维度上都证明了高斯式式定理。所有迹象都表明这在我们方法的范围内。