Mathematical Sciences: Spectral and Fractal Geometry: Analysis on Fractals, Noncommutative Geometry, and PDEs in the Fractal Domain

摘要拉皮德斯·拉皮德斯将研究“分形鼓”的振动，包括“带分形边界的鼓”和“带分形膜的鼓”。更具体地说，他将发展对分形学和具有分形界的区域的分析，以及研究和使用与康纳斯的非对易几何和相关的“量子化微积分”的新联系。长期的目标是通过将调和分析、概率论、动力系统和偏微分方程组的传统工具与非对易几何的方法相结合，更好地理解分形的精细几何结构。该项目具有许多潜在的物理和工程应用，包括在多孔介质和渗流网络中的传输、诸如微波腔的分形谐振子、相变以及在分维结构介质上的扩散。此外，它与当前的技术直接相关，如采油、计算机微芯片、雷达成像和被称为“气凝胶”的非常有效的隔热材料。

Abstract Lapidus Lapidus will study the vibrations of "fractal drums," both "drums with fractal boundary" and "drums with fractal membrane." More specifically, he will develop analysis on fractals and on regions with fractal boundary, as well as investigate and use new connections with Connes' noncommutative geometry and the associated "quantized calculus." The long term objective is gain a better understanding of the fine geometric structure of fractals by combining the traditional tools from harmonic analysis, probability theory, dynamical systems, and PDEs with methods from noncommutative geometry. This project has many potential physical and engineering applications, including transport in porous media and percolation networks, fractal resonators such as microwave cavities, phase transitions, and diffusions on fractally structured media. Further, it is directly relevant to current technology, such as oil recovery, computer microchips, radar imaging, and very efficient heat insulators called "aerogels."