Fundamental Research of Discrete Geometric Analysis and Its Applications

离散几何分析基础研究及其应用

• 批准号: 10304011
• 负责人: SUNADA Toshikazu
• 金额: $ 10.37万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A).
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10304011/
• 关键词: Discrete Laplacian; Crystal Lattice; Harper Operator; Central Limit Theorem; アルバネーゼ写像; 標準的実現; グラフの調和写像; 格子振動; 離散幾何学; グラフ; スペクトル幾何; 固有値

We have handled both geometric and analytic aspects of discrete Laplacians on infinite graphs which are main objects in discrete geometric analysis and show up in various fields of pure and applied mathematics, say the theory of discrete groups communication networks and Markov chains. Especially we obtained interesting results on large time asymptotic behaviors of transition probabilities of random walks on crystal lattices. One is the local central limit theorem, and another is asymptotic expansions. In our study, we made use of the notions of Albanese tori and Albanese maps which have the origin in algebraic geometry. In connection with this, we developed the theory of harmonic maps from graphs into Riemannian manifolds. Albanese maps is defined as a harminic maps from a finite graph into a flat torus. In this project, we have also studied the spectral properties of discrete magnetic Schroedinger. operators (Harper operators) on crystal lattices. The central limit theorem for Harper operators was established. We investigated twisted group C^* algeblas associated with Harper operators which is a generalization of non-commutative tori. As a byproduct of our research, we gave a rigorous treatment of quantized theory of lattice vibrations.

我们已经处理了无限图上离散拉普拉斯的几何和分析方面，这是离散几何分析的主要对象，出现在纯数学和应用数学的各个领域，比如离散群通信网络和马尔可夫链理论。特别是，我们在晶格上随机游动转移概率的大时间渐近行为方面得到了有趣的结果。一个是局部中心极限定理，另一个是渐近展开。在我们的研究中，我们利用了阿尔巴尼亚环面和阿尔巴尼亚映射的概念，这两个概念起源于代数几何。有鉴于此，我们将调和映射的理论从图发展到黎曼流形。阿尔巴尼亚映射被定义为从有限图到平坦环面的Harminic映射。在本项目中，我们还研究了离散磁性薛定谔的光谱性质。晶格上的算子(Harper算子)。建立了Harper算子的中心极限定理。我们研究了与非交换环面的推广Harper算子有关的扭群C^*代数。作为我们研究的副产品，我们对晶格振动的量子化理论进行了严格的处理。

项目成果

小谷元子: "Albanese maps and off-diagonal long time asymptotics for the heat kernal"Comm.Math.Phys.. 209. 633-670 (2000)

Motoko Kotani：“热核的阿尔巴尼亚图和非对角长时间渐近”Comm.Math.Phys.. 209. 633-670 (2000)

小谷元子: "Standard realizations of crystal lattices via harmonic maps"Trans A.M.S.. 353. 1-20 (2000)

Motoko Kotani：“通过调和图实现晶格的标准”Trans A.M.S.. 353. 1-20 (2000)

砂田利一: "Pressure and higher correlation functions for an Anoson diffeomorphism"Th.Ergod.& Dyn.Syst..

Toshikazu Sunada：“Anoson 微分同胚的压力和更高的相关函数”Th.Ergod.& Dyn.Syst..

小谷元子: "On asymptotics for closed geoderics in a negatively curved manifold"Math.Ann..

Motoko Kotani：“论负曲流形中闭测地线的渐近”Math.Ann..

小谷元子: "Standard realizations of crystal lattices via harmonic maps"Trans.Amer.Math.Soc..

Motoko Kotani：“通过调和图实现晶格的标准”Trans.Amer.Math.Soc..