Fractal trees, topology and geometry: analysis and applications

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Many natural systems in nature, such as trees, river networks and cardiovascular systems, are branching systems.  Fractal trees provide mathematical models for branching systems.  My research will shed new light on mathematical aspects of fractal trees, as well as develop new results for applications of fractal trees.  There are many different classes of fractal trees.  A simple class are symmetric binary fractal trees.  A tree in this class is described by a scaling ratio r and a branching angle theta.  A trunk of unit length splits into two branches, each of length r and forming an angle of theta (one to the left, one to the right).  The branching is continued ad infinitum following the same rule.  I have studied this class of trees using methods of topology.  Topology involves a type of geometry that ignores concrete spatial notions like straightness and distance, but considers properties like connectivity (holes, number of components, etc) and continuity.  Through this type of analysis, new characterizations and classifications of fractals trees continue to be developed.To date, there has been much research on natural systems and fractal dimension.  This has proved to be quite fruitful, but the fractal dimension does not fully characterize a fractal.  I am working with various collaborators in the natural sciences to find ways to use the new mathematical results in applications.There are interesting geometrical properties of fractal trees as well.  For example, there is a remarkable connection between four special trees and the famed `divine proportion'.  Thus another aspect of my research involves an investigation of other geoemtrical properties, such as tilings.Finally, with collaborators, we are exploring how to use different logics to describe and characterize fractal trees, and fractals in general.  My research program has different facets, ranging from the use of fractal tree models in the natural world to beautiful geometrical properties of the trees themselves, and will have a broad impact on the current knowledge of fractal trees.