Probabilistic aspects of growth processes

本项目致力于研究生长过程的确定性和随机元胞自动机模型，重点关注几个新的方向。第一个研究领域是长程增长过程，特别是在膨胀波边缘的行为，通过调查，取决于制度，混合时间和流体力学限制。研究了单调生长模型的成核性质和亚稳态性质，以及非单调生长模型的空间覆盖性质。利用三维介观模拟来了解雪晶的生长是一个新的研究方向。计算和分析方法的发展，以连接介观尺度动力学与微观（粒子）和宏观（连续体）版本。最后，研究者将设计组合和概率方法来理解随机环境对高维空间中重要的遗传学生长过程的影响。在广泛的科学背景下，这个项目的目的是了解各种物理系统传播干扰的原理。例如，晶体的生长一个多世纪以来一直是人们深入研究的课题，然而像雪花这样熟悉的物体的生长仍然是一个谜。研究者的研究揭示了生长过程的微观描述如何影响最终形状，其空间覆盖能力，在扩张前沿的行为以及对环境和动力学扰动的稳定性。一个重要的特征是严格的数学分析和大规模计算机实验之间的相互作用。该部分旨在为晶体生长和其他扩展过程的基础发展与物理、工程和生物学的联系，为有效的模拟和可视化算法发展与计算机科学的联系，为对基因型和表型空间结构的现实见解发展与理论遗传学的联系。随着数学和计算方法的进步，这种研究在材料科学中的晶体制造，大气科学中的降雪理解以及生物学中的遗传多样性方面具有潜在的应用前景。最后，通过吸引人的计算机图形学与公众进行交流，并为本科生和研究生提供充分的机会参与这些研究的各个方面，将普及数学及其应用。

This project is devoted to studying deterministic and random cellular automata models of growth processes, with the emphasis on several new directions. The first field of study are long-range growth processes, specifically the behavior at the edge of the expanding wave by investigating, depending on the regime, mixing times and hydrodynamics limits.The investigator also aims to study nucleation and metastability properties of monotone growth models, and space-covering properties of nonmonotone ones. A new research direction is the utilization of three-dimensional mesoscopic modeling to understand growth of snow crystals. Computational and analytical methods are developed to connect the mesoscopic scale dynamics with the microscopic (particle) and macroscopic (continuum) versions. Finally, the investigator will devise combinatorial and probabilistic methods to understand the effects of random environments on growth processes in high-dimensional spaces important in genetics.In a broad scientific context, the aim of this project is to understand principles by which various physical systems propagate disturbances. For example, crystal growth has been a subject of intense research for more than a century, yet growth of objects as familiar as snowflakes remains mysterious.The investigator's research sheds light on how microscopic description of a growth process effects the resulting shape, its space-covering ability, behavior at the expanding front, and stability to perturbations in the environment and in the dynamics. An important feature is the interplay between rigorous mathematical analysis and large--scale computer experimentation.This component aims to develop connections to physics, engineering and biology for fundamentals of crystal growth and other expansion processes, to computer science for efficient simulation and visualization algorithms, and to theoretical genetics for realistic insights into the structure of genotype and phenotype spaces. Together with the advancement of mathematical and computational methods, such research has potential applications to crystal manufacturing in material sciences, understanding snowfall in atmospheric sciences, and genetic diversity in biology. Finally, communication with general public by means of attractive computer graphics, and ample opportunities for undergraduate and graduate students' involvement in all aspects of such studies, will popularize mathematics and its applications.