Development and applications of discrete geometric analysis

离散几何分析的发展与应用

• 批准号: 21340039
• 负责人: SUNADA Toshikazu
• 金额: $ 6.74万
• 依托单位: Meiji University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2011
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21340039/
• 关键词: 離散幾何解析学; 標準的実現; 位相的結晶; 離散的代数幾何学; アーベル・ヤコビ写像; 位相敵結晶理論; 離散解析幾何学; 量子ウォーク; 結晶格子

The primarily purpose of this project was to provide a mathematical insight into the modern crystallography, a typical practical science that originated in the classification of the observed shapes of crystals. The tools we employed are adopted from algebraic topology, a field in pure mathematics cultivated during the first half of the last century. More specifically the elementary theory of covering spaces and homology is effectively used in the study of 3D networks associated with crystals. Further we formulate a minimum principle for crystals in the framework of discrete geometric analysis, which provides us with the concept of standard realizations, a canonical way to place a given crystal structure in space so as to produce the most symmetric microscopic shape. In spite of its pure-mathematical nature, this concept combined with homology theory turns out to fit with a systematic design and enumeration of crystal structures, an area of considerable scientific interest for many years. Meanwhile, standard realizations show up in asymptotic behaviors of random walks on topological crystals, the abstraction of crystal structures, and are closely related to a discrete analogue of Abel-Jacobi maps in algebraic geometry.

这个项目的主要目的是为现代晶体学提供数学见解，这是一门典型的实用科学，起源于对观察到的晶体形状的分类。我们使用的工具来自代数拓扑学，这是上个世纪上半叶发展起来的纯数学领域。更具体地说，覆盖空间和同调的基本理论有效地应用于与晶体相关的三维网络的研究。进一步，我们在离散几何分析的框架中提出了晶体的最小原理，这为我们提供了标准实现的概念，这是一种将给定晶体结构放置在空间中以产生最对称微观形状的规范方法。尽管这一概念具有纯数学的性质，但它与同调理论相结合，结果证明适合于晶体结构的系统设计和枚举，这是一个多年来备受科学关注的领域。同时，标准实现出现在拓扑晶体上随机游走的渐近行为，晶体结构的抽象，并与代数几何中Abel-Jacobi映射的离散模拟密切相关。

项目成果

A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions

关于分区数渐近的 Meinardus 定理的谱模拟

• 发表时间: 2010
• 期刊: Asymptotic Analysis
• 影响因子: 1.4
• 作者: Junichi Harada;Mitsuharu Otani;T.Tokihiro;H.Awata and Y.Yamada;Toshiaki Yokoyama;利根川吉廣;T. Tate
• 通讯作者: T. Tate

Asymptotic behavior of quantum walks on the line

量子线上行走的渐近行为

• 发表时间: 2012
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: 根上生也;Kazuhiro Yokoyama;Y. Takei;Hiroshi Kokubu;Hideo Tamura;T. Sunada and T. Tate
• 通讯作者: T. Sunada and T. Tate

Spectral structure of the Laplacian on a covering graph

覆盖图上拉普拉斯算子的谱结构

• 发表时间: 2009
• 期刊: European Journal of Combinatorics
• 影响因子: 1
• 作者: J. Harada;S. Hashimoto and M. Otani;根上生也;Y.Takei;Hiroshi Kokubu;田村 英男;Tetsuji Tokihiro;前園宜彦;Y.Yamada;Y. Higuchi and Y. Nomura
• 通讯作者: Y. Higuchi and Y. Nomura

New Metallic Carbon Crystal

• DOI: 10.1103/physrevlett.102.055703
• 发表时间: 2009-02-06
• 期刊: PHYSICAL REVIEW LETTERS
• 影响因子: 8.6
• 作者: Itoh, Masahiro;Kotani, Motoko;Adschiri, Tadafumi
• 通讯作者: Adschiri, Tadafumi

Lecture on topological crystallography

• DOI: 10.1007/s11537-012-1144-4
• 发表时间: 2012-03-01
• 期刊: JAPANESE JOURNAL OF MATHEMATICS
• 影响因子: 1.5
• 作者: Sunada, Toshikazu