Noncomutative Discrete Geometric Analysis

非计算离散几何分析

基本信息

批准号：
18540221
负责人：
SUNADA Toshikazu
金额：
$ 2.57万
依托单位：
Meiji University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

项目摘要

Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. As indicated by the title, this field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". Edges in a graph as 1-dimensional objects, however, do not play a substantial role except when we discuss geometry of graphs. Therefore our view is different from, for instance, the case of quantum graphs where differential operators on edges are vital. Actually the role of edges in analysis is just to give a neighboring relation among vertices, and difference operators linked with this relation replace differential operators. We thus do not need to worry about irregularity of functions which, for differential operators, may cause some trouble, if not serious. Instead, the combinatorial aspect of graphs creates a different kind of technical complication. Furthermore, analysis on both infinite graphs and non-compact manifold … More s involves much the same degree of difficulty.The "protagonist" in our research is a discrete analogue of Laplacians on manifolds (and related operators), which appears in many parts of mathematical sciences. In the nature of things, ideas cultivated in global analysis provide us a good guiding principle on the one hand, and the practical motivation is the moving force of theoretical progress on the other..Discrete Laplacians appear in both geometric crystallography and probability.In our research, we applied a remarkable result in the theory of random walks on crystal lattices to pin down a diamond crystal. It is interesting to rephrase the result such as "A random walker on a crystal lattice may detect the most natural way for his crystal lattice to sit in space". A diamond twin means a hypothetical crystal which shares the property of symmetry satisfied by the diamond crystal. Our result claims that there is only diamond twin, which is obtained as the standard realization of the maximal abelian covering graph of the complete graph K_4.As for the activity, I was a member of organizers of the special project held at Newton Institute, Cambridge University which stated from January and ended up in June, 2007. Less

离散的几何分析是几个传统学科的混合领域：图理论，几何，离散群体理论和概率。如标题所示，该字段涉及图形上的分析，这是“一维单元复合物”的代名词。但是，除非我们讨论图几何形状，否则图作为1维对象中的边缘没有起作用。因此，我们的观点与例如量子图的情况不同，在量子图的情况下，边缘上的差分运算符至关重要。实际上，边缘在分析中的作用只是在顶点之间提供邻近的关系，而与此关系相关的差异操作员替换了差异操作员。因此，我们不需要担心功能的不规则性，对于差异操作员而言，如果不严重，可能会造成一些麻烦。取而代之的是，图的组合方面产生了另一种技术并发症。此外，对无限图和非紧密歧管的分析……更多涉及的难度差不多。我们的研究中的“主角”是Laplacians在歧管（及相关操作员）上的离散类似物，这些类似物出现在数学科学的许多部分。在事物的本质上，一方面，全球分析中培养的想法为我们提供了一个良好的指导原则，实际动机是理论上进步的动力。重塑结果很有趣，例如“晶格上的随机步行者可能会发现他的水晶晶格坐在太空中的最自然方式”。钻石双胞胎是指一个假设的晶体，该晶体具有由钻石晶体满足的对称性的特性。我们的结果声称，只有钻石双胞胎，这是作为该活动的全图K_4的最大Abelian覆盖图获得的标准实现，我是剑桥大学牛顿学院在牛顿学院举行的特别项目的成员，该项目从一月开始，最终于2007年6月出现。

项目成果

期刊论文数量（0）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Mathematical theory of lattice vibrations

晶格振动的数学理论

DOI：
发表时间：
2006
期刊：
Pure and Appl. Math. Quaterly 2
影响因子：
0
作者：
Shubin;T.Sunada
通讯作者：
T.Sunada

On the K_4crystal

在K_4水晶上

DOI：
发表时间：
2007
期刊：
影响因子：
0
作者：
M.Kotani;T.Sunada;T.Sunada
通讯作者：
T.Sunada

Crystals that nature might miss creating

大自然可能错过的晶体

DOI：
发表时间：
2008
期刊：
Notices Amer. Math. Soc. 55
影响因子：
0
作者：
Mario Roy;Hiroki Sumi;Mariusz Urbanski;H. Sumi;R. Stankewitz and H. Sumi;H. Sumi;H. Sumi;R. Stankewitz and H. Sumi;H. Sumi;H. Sumi;角大輝;H. Sumi and M. Urbanski;H. Sumi and M. Urbanski;角大輝;H. Sumi;H. Sumi;H. Sumi;Hiroki Sumi;H. Sumi;角大輝;角大輝;H. Sumi;角大輝;角大輝;H. Sumi;角大輝;角大輝;角大輝;角大輝;H. Sumi;T. Sunada;T. Sunada
通讯作者：
T. Sunada

Large deviation and the tangent cone at infinity of a crystal lattice