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。分形树为分支系统提供了数学模型。我的研究将为分形树的数学方面提供新的亮点，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。一个简单的类是对称二叉分形树。这门课中的树是用缩放比r和分支角来描述的。一个单位长度的树干分成两个分支，每个分支的长度为r，形成一个角（一个在左边，一个在右边）。分支按照同样的规则无限延续。我用拓扑学的方法研究了这类树。拓扑包含一种几何类型，它忽略了诸如直线和距离等具体的空间概念，但考虑了诸如连接性（孔，组件数量等）和连续性等属性。通过这种类型的分析，新的特征和分形树的分类不断发展。迄今为止，对自然系统和分形维数进行了大量的研究。这已被证明是相当富有成效的，但分形维数并不能完全表征分形。我正在与自然科学领域的各种合作者合作，寻找在应用中使用新的数学结果的方法。分形树还有一些有趣的几何性质。例如，四棵特殊的树与著名的“神圣比例”之间有着显著的联系。因此，我研究的另一个方面涉及到其他几何性质的调查，如瓷砖。最后，与合作者，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。我的研究项目有不同的方面，从自然世界中分形树模型的使用到树木本身美丽的几何特性，并将对当前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.