Fractal Geometry and Dynamical Systems, with Applications

分形几何和动力系统及其应用

• 批准号: 1107750
• 负责人: Michel Lapidus
• 金额: $ 16.5万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1107750&HistoricalAwards=false
• 关键词: Fractal; Geometry; Dynamical; Systems; Applications

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.

在这个项目中，首席研究员和他的合作者(包括他的研究生和博士后)计划继续研究分形鼓的几何、动力学和光谱，既有带分形边界的鼓，也有带分形膜的鼓(分析和偏微分方程组脱离或关于分形学)。这就需要进一步发展复维理论，并研究(近似)自相似分维、多重分维和相关的非线性动力系统的潜在振荡现象。这也需要在非对易几何和整体分析中发展和使用新的工具，以研究测地线流动和测地度量的‘分形流形’。从两个主要的角度来看，通过分形学和动力学系统之间的相互作用，在整个项目中提供了一条连接线：(I)使用(多变量)复动力学，以便通过对经典抽取方法的推广，获得对某些分形鼓的光谱的详细了解。(Ii)发展了一种分形台球理论，并(在较长时期内)建立了相应的轨迹公式，将几何形状(如测地线流动)与相关分形鼓的频谱联系起来。在这种情况下，计算机辅助实验和物理实验的解释在形成适当的直觉方面起着重要的作用。这个项目旨在解决以下问题：一个人能听到分形鼓的形状吗？也就是，一个人可以通过让它振动并只听产生的声音来恢复关于一个粗略或复杂形状的几何轮廓的多少信息？这个问题，即使在光滑的欧几里得(或黎曼)几何的经典背景下，在当代数学中也发挥着核心作用。分形图是自然界中存在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。了解波如何通过这些分形体传播或光(和电磁辐射)如何反射到这些分形体上或反射出来，是一个关键的科学和数学问题。这项工作的潜在应用涉及多个领域，包括高科技(例如，用于手机技术的计算机微芯片、计算机网络和分形天线)、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(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(非常有效的声和热绝缘体，化学反应中的催化剂)。这位首席研究员之前的NSF资助的研究已经对这一领域(包括数学、物理和其他科学)产生了重大影响，他计划通过继续指导他的许多研究生和博士后，以及有前途和有创造力的本科生，继续他的研究和教育活动的广泛整合。在许多科学场馆和暑期学校讲课，创建、教授和监督与这一研究领域有关的新课程和研讨会，以及撰写与该领域有关的研究和教育书籍和科学论文。