Randomness and Geometric Structures

对于四个主题中的第一个，主要目标是理解典型的球体，其中“球体”在这里指的是在拓扑上是一个球体的二维黎曼流形，而“典型”指的是根据规范的“均匀”概率度量选择的。方法是考虑对离散结构的措施的限制，其中一致性的概念是有意义的。这位合作者打算为n个顶点上均匀选择的平面图诱导的图度量构造一个极限度量。实验和启发式证据表明，限制措施将具有连续统树的结构。对于第二个主题，合作者打算在黎曼流形上布朗运动的背景下研究图上随机游动与几何之间的关系的前人工作的含义。第三个主题涉及随机Loewner演化，由Schramm提出，作为渗流和环擦除行走的猜想标度极限。关于这一过程的这些猜想和其他几个猜想已经被证明，还有其他一些猜想仍然是开放的。第四个主题是小组概率。这位合作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不止一个数学领域，增加了领域之间的理解。共同的动机是更好地理解基本的数学对象，这些对象通常具有实用价值。具体地说，第二部分涉及图上的随机游动，这是一个在设计有效的计算机算法中应用的主题。第四部分涉及分组随机性，这是一个在电信行业中成功地用于加密和纠错的主题。

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.