Probabilistic aspects of growth processes

这个项目致力于研究生长过程的确定性和随机细胞自动机模型，重点是几个新的方向。第一个研究领域是长程生长过程，特别是通过研究扩展波边缘的行为，取决于区域、混合时间和流体动力学极限。研究者还旨在研究单调生长模型的成核和亚稳特性，以及非单调生长模型的空间覆盖特性。一个新的研究方向是利用三维介观模型来理解雪晶的生长。发展了计算和分析方法，将介观尺度动力学与微观(粒子)和宏观(连续)形式联系起来。最后，研究人员将设计组合和概率方法，以了解随机环境对高维空间生长过程的影响，这在遗传学中具有重要意义。在广泛的科学背景下，该项目的目的是了解各种物理系统传播扰动的原理。例如，一个多世纪以来，晶体生长一直是密集研究的主题，但像雪花一样熟悉的物体的生长仍然是个谜。研究人员的研究揭示了对生长过程的微观描述如何影响结果的形状、其空间覆盖能力、扩展前沿的行为以及对环境和动力学扰动的稳定性。一个重要的特点是严格的数学分析和大规模计算机实验之间的相互作用。这一部分旨在发展与物理、工程和生物学的联系，以了解晶体生长和其他膨胀过程的基本原理，发展与计算机科学的联系，以实现高效的模拟和可视化算法，并与理论遗传学建立联系，以现实地洞察基因和表型空间的结构。随着数学和计算方法的进步，这类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遗传多样性方面具有潜在的应用。最后，通过吸引人的计算机图形与普通公众进行交流，并为本科生和研究生提供充分的机会参与这类研究的各个方面，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.