Analysis, Geometry, and Spectral Theory On or Off Fractals

派打算继续并扩大他对谱几何和分形学之间关系的研究。我们计划研究“分形鼓”的振动，既有“带分形边界的鼓”(具有非常不规则边界的开集上的拉普拉斯)，也有“带分形膜的鼓”(关于分数维本身的拉普拉斯)。提出的问题与Kac的问题密切相关：“人们能听到鼓的形状吗？”以及物理学家迈克尔·贝里将其从“平滑”领域美丽地扩展到“分形”领域。虽然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，应用于分形表面的波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，我们建议使用和推广最近由Pi和Mchiel van Frankenuysen在《分数几何与数论》研究专著[La-vF2]中广泛发展的“分形弦”(一维具有分形边界的鼓)的“复数维”理论，该理论的部分动机是PI与Carl Pomance[Lapo]和Helmut Maier[Lama]早期在分形弦的(直接和)逆谱问题和Riemann假说方面的合作，以研究在几何和光谱中发生的迷人的振荡现象。(“复数维”被定义为适当的几何Zeta函数的极点。此外，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关于分形学和具有分形界的区域的分析和谱理论，以及研究凝聚态和固态物理中具有物理意义的‘动力学’问题；例如，在研究多孔或随机介质中的机械或电子输运，以及在分形体和无序系统中的热扩散。我们还打算继续我们的数学和计算机图形辅助研究([LAPA]，[LaNRG])偏微分方程组--例如拉普拉斯方程、热方程和(线性或非线性)波动方程--在具有分形边界的区域或在分形本身上。根据吸引人的实验和物理学家伯纳德·萨波瓦尔的解释，这项工作可能有助于理解自然界中离岸结构(例如海岸线、树木和血管)的形成。从长远来看，希望在这个项目(以及PI早期的研究)中开发的工具和结果将帮助我们比目前为止更深入地探索数学和物理中出现的分形图和相关对象的精细几何结构。

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.