Fundamental Research of Discrete Geometric Analysis and Its Applications

我们处理了无限图上的离散拉普拉斯算子的几何和解析问题，无限图上的拉普拉斯算子是离散几何分析的主要对象，在纯数学和应用数学的各个领域都有出现，比如离散群理论、通信网络和马尔可夫链。特别是我们得到了晶格上随机游走跃迁概率的大时间渐近行为的有趣结果。一个是局部中心极限定理，另一个是渐近展开式。在我们的研究中，我们利用了阿尔巴尼斯环面和阿尔巴尼斯映射的概念，它们起源于代数几何。据此，我们发展了从图到黎曼流形的调和映射理论。Albanese映射被定义为从有限图到平面环面的调和映射。在本课题中，我们还研究了离散磁薛定谔的谱性质。晶格上的算子（哈珀算子）。建立了哈珀算子的中心极限定理。研究了与Harper算子相关的扭曲群C^*代数，它是非交换环面的一个推广。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。

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.