Monotone and Nonmonotone Growth Models

本项目致力于研究生长过程的确定性和随机性细胞自动机模型，重点是几个新的方向。研究的第一个领域是非单调增长模型;研究者的目的是通过概率的扩展更新，重新标度和扰动技术以及各种几何方法来研究它们的增长形状和空间覆盖特性。也正在开发的是新的分析和计算方法，更高的维度和长期的模式，并与微分和积分方程的连接。高维空间的一个重要例子是超立方体，研究人员使用组合和概率方法来理解不相关和相关随机环境对基本生长过程的影响。在广泛的科学背景下，本项目的目的是理解各种物理系统传播干扰的原理。这种增长动力学的例子包括雪花和其他晶体的增长，流行病的传播，波的传播，物种之间的竞争和遗传多样性。研究者的研究对生长过程的微观性质如何影响其空间覆盖能力，对环境和动力学中扰动的影响，以及对相互作用的维度和范围的作用，都有轻微的影响。例如，中性突变、不相容性和环境严酷性的简单数学隐喻有助于理解遗传多样性的开始。该项目有一个重要的计算组件，并正在开发各种有效的计算机算法，用于复杂过程的模拟和可视化。最后，大学生参与复杂和随机动力学研究的充分机会有助于推广数学及其应用。

This project is devoted to studying deterministic and randomcellular automata models of growth processes, with the emphasison several new directions. The first field of study arenonmonotone growth models; the investigator aims to studytheir growth shapes and space-covering properties by extendingrenewal, rescaling and perturbation techniques from probability,as well as various geometric methods. Also being developed arenew analytic and computational approaches to higher dimensionaland long range models, and to connections with differentialand integral equations. An important example of a high-dimensionalspace is the hypercube, where the investigator usescombinatorial and probabilistic methods to understand theeffect of uncorrelated and correlated random environmentson basic growth processes.In a broad scientific context, the aim of this project isto understand principles by which various physical systemspropagate disturbances. Among instances of such growthdynamics are growth of snowflakes and other crystals, spreadof epidemics, propagation of waves, competition between species,and genetic diversification. The investigator's research shedslight on how microscopic properties of a growth process influenceits space-covering ability, on the effect of perturbations inthe environment and in the dynamics, and on the role of dimensionand range of interaction. For example, simple mathematical metaphorsfor neutral mutations, incompatibilities and harshness of theenvironment contribute to understanding the onset of geneticdiversity. The project has an essential computational componentand is developing a variety of efficient computer algorithmsfor simulation and visualization of complex processes. Finally,ample opportunities for undergraduate involvement in studyingcomplex and random dynamics help to popularize mathematics andits applications.