Analysis on Fractals

分形分析

• 批准号: 0652440
• 负责人: Robert Strichartz
• 金额: $ 32.96万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652440&HistoricalAwards=false
• 关键词: Analysis; Fractals

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.

对分形的分析是开发“粗略分析”计划的一部分，其中底层空间远非平滑。分形拥有很多结构，可以在这项任务中发挥优势。 私家侦探将继续他在这一领域的研究，与一般目标1）扩展的深度和范围的理论基本的例子，和2）扩展的广度类分形拉普拉斯。 特别是，他将调查以下一般类别的问题：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，能量和拉普拉斯算子的希尔伯特垫片，以及外近似方法（一种新的方法构建fractalLaplacians最近推出的PI）。 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真实的世界现象的工具。 然而，经典的分析假设了底层空间是平滑的。 真实的世界充满了粗糙的物体。 近年来，数学分析家试图在粗糙空间上建立微分方程理论。 分形给出了一些空间的例子，这些空间既极其粗糙，又具有大量的结构，从而允许分析理论的发展。 其中一种方法是由日本的Jun Kigami开创的，并由P.I.和他的同事 这个理论已经产生了对某些理想化例子的深刻理解，例如谢尔宾斯基垫片和相关空间。虽然这些空间过于对称，不可能出现在自然界的物体中，但它们已经出现在人造物体（天线和纳米分子）中。 该项目将继续这些关键例子的理论的数学发展，并扩大理论，以涵盖更广泛的分形类别，希望开发可用于建模自然发生的对象的工具。 该项目的一部分将涉及新兴的“实验数学”方法，即利用计算机模拟来探索数学问题，以期制定最终可能导致传统数学证明的假设。 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开发它。