Fractal Geometry and Dynamical Systems, with Applications

在这个项目中，主要研究者和他的合作者（包括他的研究生和博士后）计划对分形鼓的几何、动力学和光谱进行研究，包括分形边界鼓和分形膜鼓（分形分析和偏微分方程）。这需要进一步发展复杂维度理论，以及对（近似）自相似分形、多重分形和相关非线性动力系统的潜在振荡现象的研究。这也需要在非交换几何和全局分析中开发和使用新工具，以研究“分形流形”的测地线流和测地线度量。分形几何和动力系统之间的相互作用为整个项目提供了一个连接线索，从两个主要角度：(i)使用（多变量）复杂动力学，通过经典抽取方法的推广，以获得对某些分形鼓的光谱的详细了解。（ii）发展了分形台球理论，并（从长远来看）建立了相应的轨迹公式，将几何（例如测地线流）和相关分形鼓的光谱联系起来。在这种情况下，计算机辅助实验以及物理实验的解释在形成适当的直觉方面发挥着重要作用。这个项目旨在解决以下问题：“一个人能听到一个分形鼓的形状吗？”也就是说，一个人可以通过振动和倾听产生的声音来恢复多少关于粗糙或复杂形状的几何轮廓的信息？这个问题，即使在光滑的经典背景下，欧几里得（或黎曼）几何，在当代数学中也起着核心作用。分形是自然界中出现的许多复杂形状的数学理想化，如海岸线、河床、树木、计算机网络、血管网络、肺、矿石和石油分布等。理解波如何通过这些分形（或“流形”）传播，或者光（和电磁辐射）如何反射或离开它们，是一个关键的科学和数学问题。这项工作的潜在应用涉及各种领域，包括高科技（例如，计算机微芯片、计算机网络和用于手机技术的分形天线）、数学生物学和医学（癌症研究、血液循环）、地质学、应用和理论物理学（微波腔、量子引力模型中使用的随机表面）、天文学（宇宙的大尺度结构）和工程学（非常有效的隔音和隔热体）。化学反应中的催化剂)。首席研究员，他以前的NSF支持的研究已经在这个领域产生了重大影响（包括数学、物理和其他科学），他计划通过继续指导他的许多研究生和博士后，以及有前途和创造性的本科生，继续他的研究和教育活动的广泛整合。在许多科学场所和暑期学校授课，与该研究领域相关的新课程和研讨会的创建、教学和监督，以及与该领域相关的研究和教育书籍和科学论文的撰写。

In this project, the principal investigator and his collaborators (including his graduate students and postdocs) plan to pursue the investigation of the geometry, dynamics and spectra of fractal drums, both drums with fractal boundary and drums with fractal membrane (analysis and PDEs off or on fractals). This entails the further development of the theory of complex dimensions, along with the investigation of the underlying oscillatory phenomena, of (approximately) self-similar fractals, multifractals, and associated nonlinear dynamical systems. This also entails the development and the use of new tools in noncommutative geometry and global analysis in order to study the geodesic flow and the geodesic metric of 'fractal manifolds'. A connecting thread throughout this project is provided by the interplay between fractal geometry and dynamical systems, from two main perspectives: (i) the use of (multivariable) complex dynamics in order to obtain a detailed understanding of the spectra of certain fractal drums, via a generalization of the classic decimation method. (ii) The development of a theory of fractal billiards and (in the longer term) of corresponding trace formulas connecting the geometry (e.g., the geodesic flow) and spectra of the associated fractal drums. Computer-aided experiments as well as the interpretation of physical experiments play a significant role in forming appropriate intuition in this context.This projects aims at addressing the following question: "Can one hear the shape of a fractal drum" That is, how much information can one recover about the geometric contours of a rough or complex shape by making it vibrate and just listening to the resulting sounds? This question, even in the classical setting of smooth, Euclidean (or Riemannian) geometry, plays a central role in contemporary mathematics. Fractals are mathematical idealizations of many complex shapes occurring in nature, such as coastlines, river beds, trees, computer networks, networks of blood vessels, lungs, ore and oil distribution, etc. Understanding how waves propagate through these fractal shapes (or 'manifolds') or how light (and electromagnetic radiation) reflect on or off them, is a key scientific and mathematical problem. Potential applications of this work involve a variety of domains, including high technology (e.g, computer microchips, computer networks, and fractal antenna for use in cell phone technology), mathematical biology and medicine (cancer research, blood circulation), geology, applied and theoretical physics (microwaves cavities, random surfaces of use in models of quantum gravity), astronomy (large-scale structure of the universe), and engineering (very efficient sound and heat insulators, catalysts in chemical reactions). The principal investigator, whose previous NSF supported research has already had a significant impact on this area (both in mathematics, physics and other sciences), plans to continue his broad integration of research and educational activities, via the continued mentoring of his many graduate students and postdocs, as well as of promising and creative undergraduate students. lecturing in many scientific venues and summer schools, the creation, teaching and supervision of new courses and seminars connected with this research area, as well as the writing of research and educational books and scientific papers connected with this field.