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 nonmonotone growth models; the investigator aims to study their growth shapes and space-covering properties by extending renewal, rescaling and perturbation techniques from probability, as well as various geometric methods. Also being developed are new analytic and computational approaches to higher dimensional and long range models, and to connections with differential and integral equations. An important example of a high-dimensional space is the hypercube, where the investigator uses combinatorial and probabilistic methods to understand the effect of uncorrelated and correlated random environments on basic growth processes.
In a broad scientific context, the aim of this project is to understand principles by which various physical systems propagate disturbances. Among instances of such growth dynamics are growth of snowflakes and other crystals, spread of epidemics, propagation of waves, competition between species, and genetic diversification. The investigator's research sheds light on how microscopic properties of a growth process influence its space-covering ability, on the effect of perturbations in the environment and in the dynamics, and on the role of dimension and range of interaction. For example, simple mathematical metaphors for neutral mutations, incompatibilities and harshness of the environment contribute to understanding the onset of genetic diversity. The project has an essential computational component and is developing a variety of efficient computer algorithms for simulation and visualization of complex processes. Finally, ample opportunities for undergraduate involvement in studying complex and random dynamics help to popularize mathematics and its applications.
