Analysis on Fractals

分形分析是开发“粗糙分析”程序的一部分，其中底层空间远非光滑。分形具有很多结构，可以在这项任务中发挥优势。P.I.将继续他在这一领域的研究，其总体目标是1)扩展基本例子理论的深度和范围，2)扩展分形拉普拉斯类的广度。特别是，他将研究以下一般范畴的问题：分布理论、微分方程、量子力学、能量拉普拉斯算子、拉普拉斯算子的谱、希尔伯特垫片上的能量和拉普拉斯算子，以及外逼近法（一种构造分形拉普拉斯算子的新方法，最近由P.I.引入）。一些研究将涉及与本科生（主要是REU学生）合作进行的“实验数学”。数学分析为科学家提供了模拟现实世界现象的工具。然而，经典分析做出了一个默认的假设，即底层空间是光滑的。现实世界充满了粗糙的物体。近年来，数学分析人员试图在粗糙空间上构造微分方程理论。分形给出了空间的例子，这些空间既非常粗糙，又有大量的结构，允许分析理论的发展。其中一种方法是由日本的木上淳（Jun Kigami）首创的，并经过私家侦探和他的同事们的大力发展。这一理论对某些理想化的例子产生了深刻的理解，如谢尔宾斯基垫圈和相关空间。尽管这些空间太过对称，不可能出现在自然世界的物体中，但它们已经出现在人造物体中（天线和纳米分子）。该项目将继续对这些关键例子的理论进行数学发展，并将理论扩展到更广泛的分形类，希望开发出可用于自然发生对象建模的工具。该项目的一部分内容将涉及新兴的“实验数学”方法论，即使用计算机模拟来探索数学问题，以期形成可能最终导致传统数学证明的猜想。私家侦探在过去使用这种方法非常成功，并将继续与本科生合作发展它。

Analysis on fractals is part of a program to develop "rough analysis", where the underlying space is far from smooth. Fractals possess a lot of structure that can be used to advantage in this task. The P.I. will continue his research in this area, with the general goals 1) to extend the depth and scope of the theory for basic examples, and 2) to extend the breadth of the class of fractal Laplacians. In particular, he will investigate problems in the following general categories: distribution theory, differential equations, quantum mechanics, the energy Laplacian, spectra of Laplacians, energy and Laplacians on the Hilbert gasket, and the method of outer approximation (a new method of constructing fractalLaplacians recently introduced by the P.I.). Some of the research will involve "experimental mathematics" to be carried out in collaboration with undergraduate students (mainly REU students). Mathematical analysis provides scientists with the tools to model real world phenomena. However, classical analysis makes the tacit assumption that the underlying space is smooth. The real world is filled with rough objects. In recent years, mathematical analysts have attempted to construct theories of differential equations on rough spaces. Fractals give examples of spaces that are both extremely rough and yet have a great deal of structure that allows the development of an analytic theory. One approach was pioneered by Jun Kigami in Japan and intensely developed by the P.I. and his colleagues. This theory has produced a deep understanding of certain idealized examples, such as the Sierpinski gasket and related spaces. Although these spaces are far too symmetric to occur in objects in the natural word, they have already appeared in manmade objects (antennas, and nanomolecules). This project will continue the mathematical development of the theory of these key examples, and also broaden the theory to encompass wider classes of fractals, with the hope of developing tools that can be used in modeling naturally occurring objects. Part of the project will involve the emerging methodology of "experimental mathematics", in which computer simulations are used to explore mathematical questions in the hope of formulating conjectures that may eventually lead to conventional mathematical proofs. The P.I. has been very successful in using this approach in the past, and will continue to develop it in collaboration with undergraduate students.