Fractal Geometry and Applications

该项目延续并扩大了PI早期对分形几何、光谱几何和动力系统之间关系的研究。我们计划研究“分形鼓”的振动，包括“具有分形边界的鼓”（边界非常不规则的开集上的拉普拉斯算子）和“具有分形膜的鼓”（分形本身上的拉普拉斯算子）、“分形弹子”，以及相关的“复杂分形维数”（非光滑几何及其谱固有的振荡的合适度量）和“分形曲率”（分形空间中空间扭曲的合适度量）。虽然提出的理论在数学上是严谨的，但它也有自然的物理动机（例如，应用于分形表面的波散射和多孔介质的研究），并且最近（并将继续）从计算机图形学和计算机实验的使用中获得动力。我们也打算继续我们的数学和计算机图形辅助研究偏微分方程，如分形边界区域或分形本身的拉普拉斯方程、热方程和波动方程。所提出的问题与Mark Kac的问题“一个人能听到鼓的形状吗？”以及Michael Berry从“光滑”到“分形”领域的美丽扩展密切相关。预期的结果应该在凝聚态和固态物理中具有物理意义；例如，在多孔或随机介质中的机械或电传输的研究，以及在分形和无序系统中的热扩散。它们还应与地下成像和研究信息和电磁信号在崎岖地形上传播的方式有关。（最近的工程和物理应用包括新型手机、分形天线、扬声器、隔热材料、近乎最佳的隔音墙、雷达探测、催化化学反应和计算机微芯片。）这项工作可能有助于理解自然界中分形结构（如海岸线、树木和血管）的形成，以及为什么某些生物结构（如肺）既是分形的，又是实现其生物功能的近乎最佳的原因。

This project pursues and amplifies the PI's earlier investigations of the relationships between fractal geometry, spectral geometry, and dynamical systems. We plan to study the vibrations of "fractal drums," both "drums with fractal boundary" (Laplacians on open sets with very irregular boundary) and "drums with fractal membrane" (Laplacians on fractals themselves), "fractal billiards," as well as the associated "complex fractal dimensions" (a suitable measure of the oscillations intrinsic to nonsmooth geometries and their spectra) and "fractal curvatures" (a suitable measure of the way space warps in a fractal space). Although the proposed theory is mathematically rigorous, it is also naturally physically motivated (with, e.g., applications to the scattering of waves by fractal surfaces and to the study of porous media), and has recently drawn (and will continue to draw) its impetus from the use of computer graphics and computer experiments. We also intend to pursue our mathematical and computer graphics aided study of partial differential equations such as the Laplace, heat, and wave equations on regions with fractal boundary or on fractals themselves.The proposed problems are closely connected to Mark Kac's question "Can one hear the shape of a drum?" and to its beautiful extensions from the "smooth" to the "fractal" domain by Michael Berry. The expected results should be of physical significance in condensed matter and solid state physics; for example, in the study of mechanical or electrical transport in porous or in random media, as well as of heat diffusions on fractals and in disordered systems. They should also be relevant to subterranean imaging and the study of the way information and electromagnetic signals propagate across rough terrain. (Recent engineering and physical applications include new types of cell phones, fractal antennas, loudspeakers, heat insulators, nearly optimal soundproof walls, radar detection, catalytic chemical reactions, and computer microchips.) This work may help understand the formation of fractal structures (e.g. coastlines, trees and blood vessels) in nature as well as the reason why certain biological structures, such as lungs, are both fractal and nearly optimal to fulfill their biological functions.