Rigorous and Empirical Analysis of Random Cellular Automata

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9626152 Griffeath The current and planned research of David Griffeath focuses on an interplay between mathematical theory and computer visualization for complex, random and deterministic lattice dynamics. Griffeath exploits this "symbiosis" in his theoretical and empirical studies of cellular automaton dynamics and other nonlinear spatial interactions. On the theoretical side, this work is giving rise to an increasingly rich and illuminating rigorous theory of Threshold Growth, a new paradigm for discrete-time parallel updating of lattice configurations. On the empirical side, Griffeath is developing effective, interactive simulation tools for the study of complex spatial systems, designing two platforms in parallel: (a) a high-end simulator based on the new CAM8 cellular automaton machine from MIT, of which he has one of the first few prototypes in the world; and (b) a low-end 32-bit Windows 95 simulator and graphical front end called WinCA. Over the next three years, among the types of dynamical system Griffeath intends to study using this hybrid methodology are various cellular automata, interacting particle systems, and individual-based models that serve as prototypes for such complex phenomena as nucleation, spiral formation in excitable media, the shape theory of crystal growth, flocking, host-parasite interactions, dendritic growth, and phase separation by surface tension. Mathematical models for growth and competition have played a key role in the development of spatial complex systems theory since the mid 1950's. An extensive body of theory and applications, for both random and deterministic systems, has been developed, especially over the past twenty years. Modeling of such fundamental phenomena as aggregation, emergence of shape, interfacial competition, and droplet interaction sheds new light on many areas of applied science. Griffeath's work during the next funding period will be motivated especially by problems elicited from the spatial mathematical ecology community, as identi fied during an interdisciplinary conference that he organized at the Santa Fe Institute during January, 1996. He expects that the various simulation tools he is developing will be able to provide new insights into questions of genuine interest to that clientele, and that a substantial body of interesting new pure mathematics will result as well.