Spectral Invariants of Noncommutative Spaces

我提出的研究旨在扩展空间及其几何的概念，并研究这些新类型空间的曲率（弯曲）。因此，它可能对我们目前在物理学中对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，时空是一个光滑的四维流形，这一概念并不适用于量子引力，在非常小的距离和高能量下，必须用量子时空来代替。目前还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是最有希望的。另一方面，数学的内在需要也提出了这些新的空间及其几何，称为非交换几何（NCG），这仅仅是因为我们需要用几何、分析和拓扑学的工具来处理由叶理、分形的叶子构成的高度奇异的空间，或由好空间的坏子构成的空间。 数学史也是空间概念的演变史！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、盖尔方、格罗滕迪克、康纳斯和孔采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，现在又扩展到NC维。这个建议是最新的，就曲率的研究而言，也是这个佐贺中最新的一章。在数学上，我是在赫尔曼·外尔（Hermann Weyl）的工作以及卡茨（Kac）、米尔诺（Milnor）、吉尔基（Gilkey）、麦基恩-辛格（McKean-Singer）、阿蒂亚-博特（Atiyah-Bott）和康纳斯（Connes）的广泛扩展的基础上建立和扩展的，这些工作总结在谱几何的标题下，在谱几何中，人们可以听到（关于）鼓的形状的许多东西！这里的字形和鼓必须广义理解！因此，鼓可以是欧几里得空间中的一个区域，一个紧致流形，一个分形，或者，如在这个提议中，一个非交换流形。类似地，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以及它们更奇异的非对易类似物（如康纳斯）来描述的任何概念。在过去的10年里，我们已经证明，人们实际上也可以听到很多NC空间的几何形状，事实上，谱几何思想是获得有关这些新的弯曲NC空间的内在几何信息的唯一方法。除了谱几何，我现在开始将非常新的令人兴奋的想法建议随机矩阵理论和拓扑递归到研究NC空间。 我将扩展我的研究NC空间的标量和Ricci曲率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中定义类似于完全黎曼曲率的东西，并在所有偶数维中证明高斯-邦纳型定理。所有迹象都表明，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.