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.

我们已经在无限图上处理了离散的laplacians的几何和分析方面，这是离散几何分析中的主要对象，并在纯数学和应用数学的各个领域中显示，说明了离散群体通信网络和马尔可夫链的理论。尤其是我们在晶体晶格上随机行走的过渡概率的大渐近行为中获得了有趣的结果。一种是局部中央限制定理，另一个是渐近扩张。在我们的研究中，我们利用了具有代数几何形状起源的Albanese Tori和Albanese图的概念。与此相关的是，我们将谐波图理论从图形开发为Riemannian歧管。 Albanese地图定义为从有限图到平坦的圆环的Harminic图。在这个项目中，我们还研究了离散磁性Schroedinger的光谱特性。晶体晶格上的操作员（Harper操作员）。建立了Harper操作员的中心限制定理。我们研究了与Harper操作员相关的扭曲组C^*代数，这是非交通托里的概括。作为我们研究的副产品，我们对量化晶格振动理论进行了严格处理。

项目成果

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: "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)

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..

砂田利一: "Pressure and higher correlation functions for an Anosov diffeomorphism"

Toshikazu Sunada：“阿诺索夫微分同胚的压力和更高相关函数”