Growth and Motion in a Random Medium

这个项目是对数学模型的基础研究，该模型描述了在具有随机不可预测性的不规则环境中的复杂相互作用、增长和运动。其目标是发现支配这类系统的一般数学规律。这些系统在小尺度和大尺度上看起来截然不同。因此，重要的是要理解小规模进化的不同规则如何导致不同的大规模、系统范围的行为。这样的数学研究可以解释的真实世界现象包括车辆的运动，通信网络中的分组，管道中的流体颗粒，流体在多孔介质中的传播，流行病在人群中的传播，以及流体中聚合物链的波动。实验室实验证明，这些数学模型捕捉到了物理现实的基本特征。从长远来看，理解复杂的相互作用对科学和工程，进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对提案中描述的那种数学系统进行了深入而同时的研究。该项目为研究生提供了科研培训机会。本项目研究随机介质中生长和运动的数学模型。实例包括首次通过渗流、拐角生长模型、随机环境中的随机游动、定向聚合物模型和随机偏微分方程组。这项工作的目标是对这些模型的行为进行严格的数学描述，并开发用于分析它们的健壮工具。具体目标包括极限形状的正则性，最优路径的长度、起伏和几何特征等性质，遍历性质，用变分公式和熵描述大尺度极限，以及描述复杂随机几何对象的概率分布，如测地线树和竞争界面。这项工作使用的方法是严格的数学研究，辅以实验计算机模拟。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project is fundamental research on mathematical models that describe complex interactions, growth, and motion in an irregular environment with stochastic 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 communication networks, fluid particles in a tube, fluid spreading in a porous medium, epidemics advancing in a population, and the fluctuations of a polymer chain in a fluid. Laboratory experiments have demonstrated that these mathematical models capture essential features of physical reality. 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 provides research training opportunities for graduate students. 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, directed polymer models, and stochastic partial differential equations. 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, ergodic properties, 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.