Topics in geometric spectral theory

几何谱理论主题

• 批准号: 261570-2012
• 负责人: Polterovich, Iosif
• 金额: $ 2.55万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571085
• 关键词: Topics; geometric; spectral; theory

Geometric spectral theory is an area of mathematics that lies at the intersection of analysis, partial differential equations and differential geometry. The proposal focuses on the properties of eigenvalues and eigenfunctions of the Laplacian and other differential operators defined on geometric objects. Most problems are motivated by questions originating in the study of real life phenomena, such as vibration of membranes and plates, oscillations of fluids and quantum-mechanical effects. One of the problems addressed in the proposal may be stated as follows: "Can one reconstruct the shape of a water tank by observing the surface waves?" This formulation echoes the celebrated question "Can one hear the shape of a drum?" asked by M. Kac more than forty years ago. Both questions can be viewed as models of so-called inverse problems. Problems of this type arise in many important applications, such as seismic imaging in geophysics: the study of the inner structure of the earth and prediction of earthquakes using data obtained from seismic waves. Another application is electrical impedance tomography. It is a non-invasive medical imaging technique, which mathematical foundations are based on the study of the Dirichlet-to-Neumann operator. Certain properties of this operator are also addressed in the proposal. While mathematical models simplify reality in a significant way, exploring these models often provides illuminating insights and helps us better understand the corresponding physical processes. This is one of the goals of the proposed research program.

几何谱理论是一个数学领域，位于分析、偏微分方程和微分几何的交叉点。该提案重点关注拉普拉斯算子和几何对象上定义的其他微分算子的特征值和特征函数的性质。大多数问题都是由源于现实生活现象研究的问题引发的，例如膜和板的振动、流体的振荡和量子力学效应。 该提案解决的问题之一可以表述如下：“能否通过观察表面波浪来重建水箱的形状？”这个表述呼应了著名的问题“人能听到鼓的形状吗？”四十多年前，M. Kac 提出了这样的问题。 这两个问题都可以被视为所谓的逆问题的模型。 此类问题出现在许多重要应用中，例如地球物理学中的地震成像：研究地球内部结构以及使用从地震波获得的数据预测地震。另一个应用是电阻抗断层扫描。它是一种非侵入性医学成像技术，其数学基础基于狄利克雷到诺依曼算子的研究。提案中还讨论了该运算符的某些属性。 虽然数学模型在很大程度上简化了现实，但探索这些模型通常会提供启发性的见解，并帮助我们更好地理解相应的物理过程。 这是拟议研究计划的目标之一。