Analysis, Geometry, and Spectral Theory On or Off Fractals

PI打算继续和扩大他对光谱和分形几何之间关系的研究。我们计划研究“分形鼓”的振动，包括“分形边界鼓”（边界非常不规则的开集上的拉普拉斯算子）和“分形膜鼓”（分形本身上的拉普拉斯算子）。所提出的问题与卡茨的问题“人能听到鼓的形状吗？”以及物理学家迈克尔·贝里（Michael Berry）提出的从“光滑”到“分形”的美妙扩展密切相关。尽管所提出的理论在数学上是严格的，但它也有自然的物理动机（例如，分形表面对波的散射和多孔介质的研究），并且最近从计算机图形学的使用中获得了一些推动力。此外,我们建议使用和扩展“分形弦”（具有分形边界的一维鼓）的“复维”理论-最近由PI和machiel van Frankenhuysen在关于“分形几何和数论”的研究专题[La-vF2]中广泛发展，部分动机是PI早先与Carl Pomerance [LaPo]和Helmut Maier [LaMa]在分形弦和黎曼假设的（正）逆光谱问题上的联合工作-为了研究有趣的“分形边界鼓”和“分形膜鼓”在几何和频谱上出现振荡现象。(“复维”被定义为一个合适的几何zeta函数的极点。进一步，在[La-vF]中，给出了它们的结构的详细研究，例如，在自相似分形弦的情况下。我们计划进一步发展分形和分形边界区域的分析和光谱理论，并研究凝聚态和固体物理中具有物理意义的“动力学性质”问题；例如，在多孔或随机介质中的机械或电输运的研究，以及在分形和无序系统中的热扩散。我们还打算在分形边界或分形本身的区域上继续我们的偏微分方程（PDEs）的数学和计算机图形辅助研究（[LaPa], [LaNRG]），如拉普拉斯方程，热和（线性或非线性）波动方程。根据物理学家伯纳德·萨波瓦尔（Bernard Sapoval）引人入胜的实验和解释，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。从长远来看，希望这个项目（以及PI的早期调查）中开发的工具和结果将帮助我们更深入地探索分形的精细几何结构以及数学和物理学中出现的相关“物体”。

ABSTRACTThe PI intends to pursue and amplify his investigations of therelationships between spectral and fractal geometry. We plan to study thevibrations of "fractal drums", both "drums with fractal boundary"(Laplacians on open sets with very irregular boundary) and "drums withfractal membrane" (Laplacians on fractals themselves). The proposedproblems are closely connected to Kac's question "Can one hear the shape ofa drum?" and to its beautiful extensions from the "smooth" to the "fractal"domain by the physicist Michael Berry. Although the proposed theory ismathematically rigorous, it is also naturally physically motivated (with,for example, applications to the scattering of waves by fractal surfacesand the study of porous media), and has recently drawn some of its impetusfrom the use of computer graphics. Moreover, we propose to use and extendthe theory of "complex dimensions" of "fractal strings" (one-dimensionaldrums with fractal boundary)-recently developed extensively by the PI andMachiel van Frankenhuysen in the research monograph [La-vF2] on "FractalGeometry and Number Theory" and motivated in part by the PI's earlier joint work with Carl Pomerance [LaPo] and Helmut Maier [LaMa] on (direct and)inverse spectral problems for fractal strings and the Riemann hypothesis-inorder to study the fascinating oscillatory phenomena occurring in thegeometry and in the spectrum of "drums with fractal boundary" and of "drumswith fractal membrane". ("Complex dimensions" are defined as the poles ofa suitable geometric zeta function. Further, in [La-vF], a detailed studyof their structure is given, for example, in the case of self-similar fractal strings.)We plan to further develop analysis and spectral theory on fractals and onregions with fractal boundary, as well as to investigate problems of a'dynamical nature', of physical significance in condensed matter and solidstate physics; for example, in the study of mechanical or electricaltransport in porous or in random media, as well as of heat diffusions onfractals and in disordered systems. We also intend to pursue ourmathematical and computer graphics-aided study ([LaPa], [LaNRG]) of partialdifferential equations (PDEs)-such as the Laplace, heat and (linear ornonlinear) wave equations-on regions with fractal boundary or on fractalsthemselves. According to appealing experiments and interpretations by thephysicist Bernard Sapoval, this work may help understand the formation offractal structures (for example, coastlines, trees and blood vessels) innature. In the long term, it is hoped that the tools and results developedin this project (and in the PI's earlier investigations) will help us toprobe more deeply than has been so far possible the fine geometricstructure of fractals and of related 'objects' occurring in mathematicsand in physics.