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Growth and Motion in a Random Medium

随机介质中的生长和运动

基本信息

批准号：
1854619
负责人：
Timo Seppalainen
金额：
$ 33.87万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854619&HistoricalAwards=false
关键词：
Growth Motion Random Medium

项目摘要

This award supports fundamental research on mathematical models that describe complex interactions, growth and motion in an irregular environment. These mathematical systems incorporate also random unpredictability. The goal is to discover general mathematical laws that govern such systems. These systems appear quite different at small scales and large scales. So it is important to understand how different rules for small-scale evolution lead to different large-scale system-wide behavior. Real-world phenomena that such mathematical studies can illuminate include the motion of vehicles, packets in a communication network, fluid particles in a tube, wetting transitions where fluid spreads in a porous medium, epidemics advancing in a population, or the fluctuations of a polymer chain in a fluid. Over the long term understanding complex interactions has profound implications for science and engineering and thereby for society. Mathematical systems of the kind described in the proposal are intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers. This project investigates mathematical models of growth and motion in random media. Examples include first-passage percolation, the corner growth model, random walk in random environment, and directed polymer models. The objectives of this work are mathematically rigorous descriptions of the behavior of these models and the development of robust tools for their analysis. Specific goals include regularity of limit shapes, properties of optimal paths such as their length, fluctuations and geometric features, descriptions of large scale limits in terms of variational formulas and entropy, and descriptions of probability distributions of complicated random geometric objects such as trees of geodesics and competition interfaces. The methods employed in this work are those of rigorous mathematical research, aided by experimental computer simulation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该奖项支持数学模型的基础研究，这些模型描述了不规则环境中复杂的相互作用，生长和运动。这些数学系统也包含随机的不可预测性。其目标是发现支配这些系统的一般数学定律。这些系统在小尺度和大尺度上表现出很大的不同。因此，了解不同的小规模演化规则如何导致不同的大规模系统行为是很重要的。这种数学研究可以阐明的现实世界现象包括车辆的运动，通信网络中的数据包，管道中的流体颗粒，流体在多孔介质中传播的润湿转变，人口中的流行病或流体中聚合物链的波动。从长远来看，理解复杂的相互作用对科学和工程，从而对社会有着深远的影响。数学家、自然科学家、社会科学家和工程师都在同时深入地研究提案中所描述的数学系统。本项目研究随机介质中生长和运动的数学模型。例子包括第一次通过渗流，角落生长模型，随机环境中的随机行走，定向聚合物模型。这项工作的目标是数学上严格的描述这些模型的行为和强大的工具，他们的分析发展。具体目标包括极限形状的规律性，最优路径的性质，如它们的长度，波动和几何特征，大规模限制的变分公式和熵的描述，以及复杂的随机几何对象的概率分布的描述，如测地线和竞争界面的树。这项工作中采用的方法是严格的数学研究方法，并辅以实验计算机模拟。该奖项反映了NSF的法定使命，并且通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。