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.
PI's research is focused on two prominent themes. The first was to understand how simple spatial rules called cellular automata (CA) propagate disturbances across the available space. Such CA can be deterministic or random; this field of investigation is often referred to as growth models. These rules are partly based on modeling physical processes such as excitable media, crystal growth and spread of infections, partly from ideas from computer science such as complexity and artificial life, and partly from a desire to understand the connections between local and global properties of an evolving system. The second prominent thread is nontrivial computational component, which involves simulation numerical and statistical methods, development of algorithms, and visualization techniques. The CA results address three different types of evolution: replication, periodicity, and chaos, and offer a prospect for comprehensive theory of one dimensional CA started from seeds some 30 years after such dynamics were introduced by Stephen Wolfram . The PI has, with collaborators, developed a fully realistic model of snow crystal growth and collaborated with atmospheric scientists and graphic artists on their applied and aesthetic aspects. The resulting images have been used on callendars (by American Mathematical Society) and featured in gallery exbitions. PIs work on bootstrap percolation demonstrates the importance of mathematical theory in interpreting simulation results. The PI has also collaborated on studying fitness landscapes genetics to investigate effects of environemental conditions and genetic incompatibility to speacies diversity, and has, with collaborators, initiated investigation of coalition networks.
