Spectral Invariants of Noncommutative Spaces

我提议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，它可能会对我们目前对物理学中时空结构的理解，以及对数学本身的理解产生潜在的影响。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4维流形，不适合量子引力，必须在非常小的距离和很高的能量下被量子时空取代。只是还不清楚到底会有什么样的替代品！有几个提议，但我相信我们的提议是最有希望的。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)，因为我们需要用几何、分析和拓扑学的工具来处理具有叶的高度奇异的空间、分形的空间或好空间的坏商。 数学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、格尔方、格罗森迪克、康纳斯和康采维奇的工作已经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。这个提议是最新的，就曲率的研究而言，是这一传奇故事的最新篇章之一。 在数学上，我是在Hermann Weyl的工作和Kac、Milnor、Gilkey、McKean-Singer、Atiyah-Bott和Connes的大量扩展的基础上进行的，总结在谱几何的标题下，在那里人们可以听到(关于)鼓的形状的(许多事情)！在这里，字形和鼓必须被广泛理解！因此，鼓可以是欧几里得空间中的区域、紧致流形、分形域，也可以是非对易流形。类似地，形状可以理解为体积、标量曲率 Ricci曲率，以及，一般而言，任何可以根据像拉普拉斯或狄拉克及其更奇异的非对易类似物如康纳斯的算符的谱来捕捉的概念。在过去的10年里，我们能够证明的是，人们实际上也可以听到许多NC空间的几何图形，事实上，光谱几何思想是人们获得关于这些新的弯曲NC空间的内在几何图形的唯一途径。除了谱几何，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融入到NC空间的研究中。 我将把我对NC空间的标量曲率和Ricci曲率的研究扩展到高维和非共形度量。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，并证明所有偶数维上的Gauss-Bonnet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.