Monotone and Nonmonotone Growth Models

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，并强调了几个新的方向。第一个研究领域是单调生长模型；研究者的目标是通过概率的扩展更新、重标度和摄动技术以及各种几何方法来研究它们的生长形状和空间覆盖特性。同时也在开发新的分析和计算方法来研究高维和远程模型，以及与微分和积分方程的联系。高维空间的一个重要例子是超立方体，研究者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，这个项目的目的是理解各种物理系统传播干扰的原理。这种生长动力学的实例包括雪花和其他晶体的生长、流行病的传播、波浪的传播、物种之间的竞争和遗传多样化。研究者的研究揭示了生长过程的微观特性如何影响其空间覆盖能力，环境和动力学扰动的影响，以及相互作用的维度和范围的作用。例如，中性突变、不相容和严酷环境的简单数学隐喻有助于理解遗传多样性的起源。该项目具有重要的计算组件，正在开发各种高效的计算机算法，用于复杂过程的模拟和可视化。最后，本科生参与研究复杂和随机动力学的充分机会有助于普及数学及其应用。

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.