Spectral Invariants of Noncommutative Spaces

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，它可以对我们目前对物理学中时空结构的理解以及数学本身产生潜在的影响。事实上，众所周知，在爱因斯坦的广义相对论中，目前的时空概念是一个光滑的四维流形，这对于量子引力来说是不够的，必须在非常小的距离和高能量下，用量子时空来取代。目前还不清楚到底会有什么替代品！有几个建议，但我相信我们的建议最有希望。另一方面，数学的内部需求也提出了这些新的空间及其几何，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状叶的高度奇异空间、分形或良好空间的坏商。*******数学的历史也是空间概念演变的历史！从欧几里得的元素开始，笛卡尔、高斯、黎曼、希尔伯特、冯·诺伊曼、盖尔芬德、格罗腾迪克、康内斯和康茨维奇的工作，系统地将空间及其几何的概念扩展到更复杂的范式，现在扩展到NC维度。这个提议是最新的一个，就曲率的研究而言，是这个传奇故事的最新章节。******在数学上，我是建立和扩展赫尔曼·魏尔的工作，以及由卡茨、米尔诺、吉尔基、麦基恩-辛格、阿蒂亚-博特和科恩斯对其进行的广泛扩展，这些工作在光谱几何的标题下进行了总结，其中表明人们可以听到（关于）鼓的形状的许多事情！在这里，形状和鼓这两个词必须被广泛地理解！因此，鼓可以是欧几里得空间中的一个定义域，一个紧化流形，一个分形，或者，在这个提议中，一个非交换流形。同样地，形状可以被理解为体积，标量曲率，里奇曲率，一般来说，任何可以用拉普拉斯或狄拉克等算子的频谱以及它们的非交换类似物来描述的概念。在过去的10年里我们已经证明的是人们实际上也可以听到很多NC空间的几何实际上，谱几何思想是人们获得这些新的弯曲NC空间的内在几何信息的唯一途径。除了光谱几何之外，我现在开始将随机矩阵理论和拓扑递归提出的非常令人兴奋的新想法纳入NC空间的研究中。******我将把我对NC空间的标量和里奇曲率的研究扩展到更高的维度和非共形度量。这个提议的一个目标是最终在NCG中定义类似完整黎曼曲率的东西，并在所有偶数维中证明高斯-博内型定理。所有迹象表明，这是我们的方法所能达到的。* * *

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. ***