Limits via sampling of large discrete and continuous structures

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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.