Fractal trees, topology and geometry: analysis and applications

自然界中的许多自然系统，如树木、河流网络和心血管系统，都是分支系统。而分枝树为分支系统提供了数学模型。我的研究将对分形树的数学方面有新的认识，并为分形树的应用开发新的结果。分形树有许多不同的类别。简单的类别是对称的二进制分形树。这类树中的一棵树用比例r和分枝角度θ来描述。单位长度的树干分成两个分支，每个分支的长度为r，并形成一个角(从左到左，向右一)。分支是按照相同的规律无限地继续的。我曾用拓扑学的方法研究过这类树。拓扑学涉及一种几何类型，它忽略了具体的空间概念，如直线度和距离，但考虑了性质，如连通性(洞，组件的数量等)和连续性。通过这种类型的分析，新的刻画和分类的分形树继续发展。到目前为止，已经有很多关于自然系统和分维的研究。这被证明是相当有成效的，但分维并不能完全描述分形学。我正在与自然科学中的各种合作者合作，寻找在应用中使用新的数学结果的方法。分形树也有有趣的几何性质。例如，四棵特殊的树与著名的神圣比例之间存在着显著的联系。因此，我研究的另一个方面涉及到对其他大地测量性质的调查，如平铺。最后，与合作者一起，我们正在探索如何使用不同的逻辑来描述和表征分形树，以及一般的分形图。我的研究计划有不同的方面，范围从自然界中使用的分形树模型到树木本身的美丽几何特性，并将对目前对分形树的知识产生广泛的影响。

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.