Limits via sampling of large discrete and continuous structures

许多大型和复杂的系统(如通信网络)由相互连接的元件组成，它们通过对现有结构的小范围重新配置和额外元件的添加而随着时间的推移而发展。在许多数学领域和更广泛的科学领域中，对理解复杂对象特别有成效的一个观点是，对对象之间的距离提出了适当的概念，并引入了“理想”无限对象，使得有限对象随着其变大而越来越接近目标无限对象。第一个子项目调查了这样的情况，其中一个信息性的距离概念是，如果对象在统计意义上接近，则认为它们是接近的；更具体地说，子项目考虑各种类型的结构，其中存在给定大小的随机抽样子结构的明确概念，并且其中一个声明，如果从每个结构中抽样的不同大小的子结构的概率行为相似，则两个结构是接近的。这个子项目的目标是调查在这些设置和相应的随机抽样概念中是否存在“理想的”无限结构，以便从大型有限结构中抽样的子结构与从目标无限结构中抽样的子结构的行为相似。第二个子项目研究一种意义，在这种意义下，大型和复杂的对象可以被分解成更简单的构件，这些构件以规定的方式组合在一起。这种分解的一个典型例子是将整数分解为素数的乘积。这里的注意力集中在可以通过随机采样点离散到任意精度的几何对象上，整数乘法的类比是笛卡尔乘积的形成。拟议的研究的后半部分由两个应用于生物学的概率论领域的子项目组成。其中之一涉及对元基因组数据的分析，在元基因组数据中，通过对存在的所有生物体的遗传物质进行批量采样，并随后将这些生物体置于微生物生命的参考“进化家谱”上，来调查环境(例如人类肠道)的微生物多样性。这项研究将开发新的方法来了解一系列这样的样本彼此不同的方式，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。后半部分的另一个子项目涉及生态和人口动态。它试图模拟在空间和时间上不同的环境中地理上分散的种群的增长，并阐明环境条件、分散战略和对资源的竞争的影响如何相互作用来影响种群的长期生存。

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.