Limits via sampling of large discrete and continuous structures

许多大型和复杂的系统（例如，通信网络）由互连的元件组成，并且它们通过对现有结构的小的重新配置结合额外元件的添加而随时间发展。 在数学和科学的许多领域中，一个对于理解复杂对象特别富有成效的观点是制定了对象之间距离的适当概念，并引入了“理想”无限对象，使得有限对象随着它们变得越来越大而越来越接近目标无限对象。 第一个分项目调查的情况是，距离的一个信息概念是，如果物体在统计意义上很近，就认为它们很近;更具体地，该子项目考虑了各种类型的结构，其中有一个井，定义了给定大小的随机采样子结构的概念，并且一个声明两个结构是接近的，如果从它们都是相似的。 该子项目的目标是调查是否有“理想”的无限结构在这些设置和相应的随机抽样的概念，这样的子结构从大型有限结构采样的行为类似于那些从目标无限结构采样。 第二个分项目研究的是一种意义，即大型和复杂的物体可以被分解成更简单的积木，并以规定的方式组合起来。 这种分解的一个典型例子是将整数分解为素数的乘积。 这里的注意力是几何对象，可以离散到任意程度的精度采样点随机和模拟的乘法整数是形成笛卡尔products.The后半部分的拟议研究包括两个子项目在该地区的概率论应用于生物学。 其中之一涉及宏基因组数据的分析，其中环境的微生物多样性（例如，人类肠道）通过从存在的所有生物体中大量取样遗传物质并随后将生物体置于微生物生命的参考“进化家谱”上来调查。 该研究将开发新的方法来了解这些样本之间的差异以及这些差异如何受到外部因素的影响，例如某些药物的存在。 后半部分的另一个分项目涉及生态和人口动态。 它试图模拟地理上分散的人口在空间和时间上异质的环境中的增长，并阐明环境条件、分散战略和资源竞争的影响如何相互作用，影响人口的长期生存。

Many large and complex systems (e.g., communication networks) are composed of interconnected elements and they develop over time by small reconfigurations of the existing structure combined with the addition of extra elements. A point-of-view that has been particularly fruitful for understanding complex objects in many areas of mathematics, and science more generally, has been the formulation of an appropriate notion of distance between objects and the introduction of "ideal" infinite objects such that the finite objects get closer and closer to a target infinite object as they become larger. The first subproject investigates situations in which an informative notion of distance is that objects are deemed to be close if they are close in a statistical sense; more specifically, the subproject considers various classes of structures where there is a well-defined notion of a randomly sampled substructure of a given size and one declares that two structures are close if the probabilistic behaviors of substructures of various sizes sampled from each of them are similar. The goal of the subproject is to investigate whether there are "ideal" infinite structures in these settings and corresponding notions of random sampling such that the substructures sampled from large finite structures behave similarly to those sampled from a target infinite structure. The second subproject investigates one of the senses in which large and complex objects may be broken down into simpler building blocks that are combined in a prescribed manner. A prototypical instance of such a decomposition is the factorization of whole numbers as products of prime numbers. Here the attention is on geometric objects which can be discretized to an arbitrary degree of precision by sampling points at random and the analogue of the multiplication of whole numbers is the formation of Cartesian products.The latter half of the proposed research consists of two subprojects in the area of probability theory applied to biology. One of these involves the analysis of metagenomic data in which the microbial diversity of an environment (e.g., the human gut) is surveyed by bulk sampling of genetic material from all organisms present and the subsequent placement of the organisms on a reference "evolutionary family tree" of microbial life. The research will develop new methods for understanding the ways in which an array of such samples differ from each other and how these differences are affected by external factors such as the presence of certain medications. The other subproject in the latter half deals with ecology and population dynamics. It seeks to model the growth of geographically dispersing populations in an environment that is heterogeneous in space and time and shed light on questions about how environmental conditions, dispersal strategies and the effects of competition for resources interact to influence the long-term survival of the population.