Randomness and Geometric Structures

随机性和几何结构

• 批准号: 0206781
• 负责人: Daniel Stroock
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0206781&HistoricalAwards=false
• 关键词: Randomness; Geometric; Structures

For the first out of four topics, the main goal is to understand the typical sphere, where "sphere" here means a two-dimensional Riemannian manifold that is topologically a sphere, and "typical" means chosen according to a canonical "uniform" probability measure. The approach is to consider a limit of measures on discrete structures, where the concept of uniformity is meaningful. The co-investigator intends to construct a limiting measure for the graph metric induced by uniformly chosen planar graphs on n vertices. Experimental and heuristic evidence suggests that the limiting measure will have the structure of a continuum tree. For the second topic, the co-investigator intends to study the implications of previous work on the relationship between random walks and geometry on graphs in the context of Brownian motion on Riemannian manifolds. The third topic concerns the stochastic Loewner evolution, introduced by Schramm as a conjectured scaling limit for percolation and loop-erased walk. These and several other conjectures concerning this process have been proved and yet others remain open. The fourth topic is probability in groups. The co-investigator intends to continue to work on automorphism groups of regular trees, which are fundamental objects of study in the theory of p-groups. All four parts of the project involve more than one area of mathematics, increasing understanding between fields. The common motivation is to understand better fundamental mathematical objects that are often of utilitarian use. In particular, the second part concerns random walks on graphs, a topic which has been applied in the design of efficient computer algorithms. The fourth part concerns randomness in groups, a topic which has been used successfully in the telecommunications industry for encryption and error-correction.

对于四个主题中的第一个，主要目标是理解典型的球面，这里的“球面”是指拓扑上是球面的二维黎曼流形，而“典型”是指根据规范的“均匀”概率测度选择的。该方法是考虑离散结构的措施的限制，其中的概念是有意义的一致性。共同研究者打算构造一个限制措施的图度量诱导均匀选择平面图的n个顶点。实验和启发式证据表明，限制措施将有一个连续树的结构。对于第二个主题，合作研究者打算研究以前的工作的影响随机游动和几何图形的布朗运动的上下文中的黎曼流形之间的关系。第三个主题是关于随机Loewner演化，介绍了Schramm作为一个约束的缩放限制渗流和循环擦除行走。关于这一过程的这些和其他几个假设已经得到证明，但其他假设仍然是开放的。第四个主题是群体中的概率。共同研究者打算继续研究正则树的自同构群，这是p群理论的基本研究对象。该项目的所有四个部分都涉及一个以上的数学领域，增加了领域之间的理解。共同的动机是为了更好地理解通常具有实用性的基本数学对象。特别是，第二部分涉及图上的随机游动，这是一个已被应用于设计有效的计算机算法的主题。第四部分涉及分组的随机性，这是一个在电信行业成功用于加密和纠错的主题。