Dynamical Riemannian Geometry of Natural Systems

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Geometry plays an important role in understanding many natural processes. This proposal will study how Riemannian geometry contributes to the evolution of two different dynamical systems through the construction of mathematical models which are used to provide the basis for computer simulations. (1) A new mathematical model of plant growth has been constructed that couples the changes in the geometry of plant to the transport and deposition of the material required to make them increase in size and change morphology. The model has been tested in the case of the growth of blades of grass and corn roots with good agreement between the numerical simulations and the observations. The extension to two-dimensional objects will compare measurements made on the growth of tobacco and other leaves to determine what genetic and what environmental effects govern the growth and formation of structure in plant leaves and petals. In addition to performing the numerical simulations, the use of computer graphic techniques will be applied to produce a visualization of the growth process. (2) The formation of black holes having electric charge will be studied using the Einstein equations of general relativity. There are a number of issues that have yet to be fully resolved concerning the formation and stability of charged black holes. First it is known that the interior of a pre-existing charged black hole becomes unstable if matter or fields fall into it. The question of whether the structure of the pre-existing black hole is compatible with one that forms from the realistic collapse of a charge fluid will be studied using the methods of numerical relativity. The second question concerns the nature of the the collapse when it is at the boundary between collapse to a black hole and the formation of a spacetime without a black hole. These simulations will be the first to study as much of the complete spacetime as is possible.