Fractal Geometry and Applications

这个项目追求和扩大PI的早期调查分形几何，光谱几何和动力系统之间的关系。我们计划研究“分形鼓”的振动，这两种鼓都是“具有分形边界的鼓”。（Laplacian在具有非常不规则边界的开集上）和“具有分形膜的鼓”（Laplacians对分形本身），“分形台球，以及相关的“复杂分形维数”（非光滑几何及其光谱固有振荡的适当度量）和“分形曲率”（分形空间中空间扭曲方式的适当度量）。虽然所提出的理论在数学上是严格的，但它也是自然的物理动机（例如，应用于分形表面的波的散射和多孔介质的研究），最近已经（并将继续）从计算机图形学和计算机实验的使用中获得推动力。我们还打算继续我们的数学和计算机图形辅助研究的偏微分方程，如拉普拉斯，热和波动方程的区域与分形边界或分形本身。提出的问题是密切相关的马克Kac的问题“一个人能听到鼓的形状？以及迈克尔·贝里从“光滑”域到“分形”域的美丽扩展。预期的结果应该在凝聚态和固态物理学中具有物理意义;例如，在多孔或随机介质中的机械或电传输的研究中，以及在分形和无序系统中的热扩散。它们还应该与地下成像以及信息和电磁信号在粗糙地形中传播方式的研究有关。（最近的工程和物理应用包括新型手机，分形天线，扬声器，隔热体，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（例如海岸线，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，也是实现其生物功能的最佳选择。

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.