Scalar curvature, spectral zeta functions and local geometric invariants for noncommutative spaces

非交换空间的标量曲率、谱 zeta 函数和局部几何不变量

基本信息

  • 批准号:
    RGPIN-2014-04087
  • 负责人:
  • 金额:
    $ 1.68万
  • 依托单位:
  • 依托单位国家:
    加拿大
  • 项目类别:
    Discovery Grants Program - Individual
  • 财政年份:
    2015
  • 资助国家:
    加拿大
  • 起止时间:
    2015-01-01 至 2016-12-31
  • 项目状态:
    已结题

项目摘要

Geometry is about measurement of shapes, and spaces in general. Through classical differential geometry we have learned how to measure distances and volumes, as well as the curvature of a given space in all dimensions. How to define and compute the curvature of a noncommutative space? A noncommutative space is a much more complicated modern analogue of a classical space. Chaotic and fuzzy character of these new types of spaces, specially lack of classical points, renders almost all of the classical methods useless. Ideas from spectral geometry and quantum mechanics, namely information about a space encoded in the spectrum of its natural geometric operators like Dirac and Laplacian gives a clue as to how to proceed in the NC case. The celebrated Weyl's law on the asymptotic distribution of eigenvalues of the Hodge-de Rham Laplacian of a closed Riemannian manifold in terms of its volume is the first result of this kind. In general the short time asymptotic expansion of the trace of the heat kernel gives an infinite sequence of spectral invariants. Borrowing words of Marc Kac: one can hear the dimension, volume and scalar curvature. The essence of this situation is axiomatized by Alain Connes in noncommutative geometry (NCG) under the concept of spectral triple. With enough regularity condition spectral triples can be thought of as noncommutative spin Riemannian manifolds. Before 2010, no real computation of curvature in a curved NC space was known or seemed feasible. For the first time after more than 30 years into NCG, in the past 4 years we (and independently and simulataneously Connes and Moscovici) were able to obtain a formula for the curvature of a curved NC 2-d torus. We (Fathizadeh-Khalkhali) have also obtained a formula for the scalar curvature of a NC curved 4-d torus. These are formidable formulas which in no way can be obtained by deforming the classical curvature formulas. This is achieved by evaluating the value of the (analytic continuation of the) spectral zeta functional \zeta_a(s) := Trace(a \Delta-s) at s = 0. A new purely noncommutative feature here is the appearance of the modular automorphism group from the theory of type III von Neumann factors and quantum statistical mechanics in the final formula for curvature. A byproduct is a Gauss-Bonnet theorem for NC 2-d torus. Other tools like Connes' trace theorem, and a noncommutative Wodzicki residue has been also obtained by us. A totally fresh and unchartered territory is now opened with so many open and interesting fundamental problems waiting to be studied. I am planning to build upon the breakthroughs I had in the last 4 years and continue my research in understanding the curved geometry of noncommutative spaces. This includes: Constructing new noncommutative Riemannian manifolds, finding the NC Gauss-Bonnet density in dimension 4, computing the scalar curvature and Gauss-Bonnet theorem of noncommutative toroidal orbifolds, extending my Riemann-Roch theorem to all holomorphic line bundles on noncommutative 2-torus, verifying Chamseddine-Connes conjectures for spectral action for Robertson-Walker metrics, proving the conformal invariance of the eta invariant for noncommutative 3-torus, conceptual understanding of our curvature formula (it is extremely important to understand at a more conceptual level the amazing cancellations that occur in our calculations with noncommutative pseudodifferential symbols, and why at the end thousands of terms cancel), establishing higher order corrections to our Weyl's law for NC curved tori (analogue of Hormander's celebrated theorem), finding new examples of noncommutative Einstein manifolds, study of quantum Yang-Mills theory on noncommutative 4-torus.
几何学是关于形状和空间的测量。通过经典微分几何,我们学会了如何测量距离和体积,以及给定空间在所有维度上的曲率。如何定义和计算非交换空间的曲率?非交换空间是对经典空间更为复杂的现代类比。这些新类型空间的混沌性和模糊性,特别是缺乏经典点,使得几乎所有的经典方法都无效。来自谱几何和量子力学的思想,即编码在自然几何算符(如狄拉克和拉普拉斯算符)谱中的空间信息,为如何在NC情况下进行提供了线索。著名的关于封闭黎曼流形的Hodge-de Rham拉普拉斯特征值在体积上的渐近分布的Weyl定律是这类的第一个结果。一般地,热核轨迹的短时间渐近展开给出了谱不变量的无穷序列。借用马克·卡茨的话:人们可以听到尺寸、体积和标量曲率。 在非交换几何(NCG)中,Alain Connes在谱三重的概念下公理化了这种情况的本质。在足够正则的条件下,谱三元组可以被认为是非交换的自旋黎曼流形。在2010年之前,没有真正的曲率计算在一个弯曲的NC空间是已知的或似乎可行的。 在进入NCG 30多年后,在过去的4年里,我们(以及独立和同时的Connes和Moscovici)第一次能够获得弯曲NC的曲率公式 二维环面。我们(Fathizadeh-Khalkhali)也得到了NC弯曲四维环面的标量曲率公式。这些都是不可思议的公式,它们绝不可能通过变形经典曲率公式而得到。这是通过计算(的解析延拓)的值来实现的。 光谱zeta函数\zeta _a(s):= Trace(a \Delta -s), s = 0。这里一个新的纯非交换特性是外观 从III型冯·诺依曼因子理论和量子统计力学在曲率最终公式中的模自同构群。副产物是NC二维环面的高斯-博内定理。我们也得到了其他的工具,如Connes的迹定理和一个非交换的Wodzicki残数。一个全新的未知领域正在打开,有许多开放和有趣的基本问题等待研究。 我计划在过去4年的突破基础上继续研究非交换空间的弯曲几何。这包括: 构造新的非交换黎曼流形,求出4维的NC高斯-庞内密度,计算非交换环面轨道的标量曲率和高斯-庞内定理,将黎曼-罗奇定理推广到非交换2环面上的所有全纯线束,验证Robertson-Walker度量的谱作用的Chamseddine-Connes猜想,证明非交换3环面上的eta不变量的共形不变性,对曲率公式的概念性理解(在更概念化的层面上理解我们计算中出现的不可交换伪微分符号的惊人消去,以及为什么最后会有数千项消去,这一点非常重要),建立NC弯曲环面Weyl定律的高阶修正(类似于著名的Hormander定理), 寻找非对易爱因斯坦流形的新例子,研究非对易4环面上的量子杨-米尔斯理论。

项目成果

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Khalkhali, Masoud其他文献

Khalkhali, Masoud的其他文献

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{{ truncateString('Khalkhali, Masoud', 18)}}的其他基金

Spectral Invariants of Noncommutative Spaces
非交换空间的谱不变量
  • 批准号:
    RGPIN-2019-04748
  • 财政年份:
    2022
  • 资助金额:
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
Spectral Invariants of Noncommutative Spaces
非交换空间的谱不变量
  • 批准号:
    RGPIN-2019-04748
  • 财政年份:
    2021
  • 资助金额:
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
Spectral Invariants of Noncommutative Spaces
非交换空间的谱不变量
  • 批准号:
    RGPIN-2019-04748
  • 财政年份:
    2020
  • 资助金额:
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
Spectral Invariants of Noncommutative Spaces
非交换空间的谱不变量
  • 批准号:
    RGPIN-2019-04748
  • 财政年份:
    2019
  • 资助金额:
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
Scalar curvature, spectral zeta functions and local geometric invariants for noncommutative spaces
非交换空间的标量曲率、谱 zeta 函数和局部几何不变量
  • 批准号:
    RGPIN-2014-04087
  • 财政年份:
    2018
  • 资助金额:
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
Scalar curvature, spectral zeta functions and local geometric invariants for noncommutative spaces
非交换空间的标量曲率、谱 zeta 函数和局部几何不变量
  • 批准号:
    RGPIN-2014-04087
  • 财政年份:
    2017
  • 资助金额:
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
Scalar curvature, spectral zeta functions and local geometric invariants for noncommutative spaces
非交换空间的标量曲率、谱 zeta 函数和局部几何不变量
  • 批准号:
    RGPIN-2014-04087
  • 财政年份:
    2016
  • 资助金额:
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
Scalar curvature, spectral zeta functions and local geometric invariants for noncommutative spaces
非交换空间的标量曲率、谱 zeta 函数和局部几何不变量
  • 批准号:
    RGPIN-2014-04087
  • 财政年份:
    2014
  • 资助金额:
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
Hopf cyclic cohomology, twisted local index formula, and noncommutative complex geometry
Hopf 循环上同调、扭曲局部指数公式和非交换复几何
  • 批准号:
    184060-2009
  • 财政年份:
    2013
  • 资助金额:
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
Hopf cyclic cohomology, twisted local index formula, and noncommutative complex geometry
Hopf 循环上同调、扭曲局部指数公式和非交换复几何
  • 批准号:
    184060-2009
  • 财政年份:
    2012
  • 资助金额:
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual

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  • 批准号:
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Scalar curvature, spectral zeta functions and local geometric invariants for noncommutative spaces
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非交换空间的标量曲率、谱 zeta 函数和局部几何不变量
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曲率流和谱估计。
  • 批准号:
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非交换空间的标量曲率、谱 zeta 函数和局部几何不变量
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