Proposal: DMS-9800714 Principal Investigator: Nikolai Makarov
Abstract: The goal of the project is to advance knowledge in some well-known areas of complex analysis, including the problems of conformal welding, diffusion limited aggregation, and the ergodic theory of harmonic measure on Julia sets.
Conformal mappings and harmonic measure are major and classical tools in the study of two-dimensional objects. They have been used in science and engineering for many years. Recently, there has been considerable interest in the extension of the classical theory to some classes of sets which have complicated ("fractal", "chaotic") structure and typically arise in dynamical systems or as random objects. Some of the most interesting examples include so-called "Laplacian clusters," certain random aggregates whose rate of growth depends on harmonic measure. Laplacian clusters serve as mathematical models for a large number of natural phenomena such as the process of diffusion limited aggregation, which has surfaced recently as a model for the formation of crystals, viscous fingering, and dielectric breakdown. Laplacian clusters have been extensively studied in the physical literature. The project will explore this topic, along with some other basic open problems concerning the general theory of conformal mappings and harmonic measure, as well as applications to dynamical and random fractals.
