Isoperimetry and spectral geometry

等周测量和光谱几何

• 批准号: RGPIN-2022-04247
• 负责人: Girouard, Alexandre
• 金额: $ 2.7万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759463
• 关键词: Isoperimetry; spectral; geometry

Vibrations and quantum mechanical effects are ubiquitous in science, in technology and in everyday life, from the design of musical instruments to nanotechnology and stability of planes. Mathematics provide the adequate language to describe these phenomena: the natural frequencies of a vibrating structure and the energy levels of quantum systems are both modeled by eigenvalues of operators that act on various spaces, such as surfaces, manifolds, graphs and even fractals. Spectral geometry is the study of the interplay between the eigenvalues of an operator and the geometry of the space on which it is defined.  A fruitful approach to understanding the geometry of various spaces is through the investigation of its isoperimetric properties. This is a classical topic going back to antiquity: among all plane figures of prescribed area, circles have the shortest perimeter. In its modern incarnation, similar problems are asked and solved for various geometric and physical quantities: what shape should a solid have to minimize the heat loss through its boundary? What shape should the skin of a drum have so that its lowest pitch be the gravest possible? The long-term aim of my research is to develop a deep understanding of the isoperimetric properties of the eigenvalues of Laplace and Dirichlet-to-Neumann (DtN) operators. Hand in hand with Fourier theory, Laplace operators are used throughout the sciences to model random motion, heat transmission, wave propagation and light.  Despite not being as well known, the DtN operator is particularly interesting. Imagine that an electric potential is applied at the surface of a solid body. The resulting current flux across its surface depends on the interior conductivity of the body. Recovering the conductivity inside the body from measurements at the surface is known as the Calderón problem. Mathematically, the voltage-to-current operator is the DtN operator. It is useful in medical imaging and in geophysical prospection. The spectral properties of the DtN operator have recently found applications in shape analysis and computational brain science. In my work I use tools from Riemannian geometry, discretization theory and coarse geometry to probe isoperimetric-type properties of eigenvalues of these Laplace and DtN operators.  Recently, I have started studying the variational eigenvalues associated to Radon measures. This leads to the unification of several eigenvalue problems, previously thought to be completely distinct. For instance the eigenvalues of the DtN operator and of weighted Laplace operators are instances of these variational eigenvalues. The proposed research will explore continuity and limit properties of these eigenvalues, in particular for family of measures that become singular. Some of our goals are to obtain sharp isoperimetric-type bounds for eigenvalues of spaces of arbitrary dimension, and to understand spectral asymptotics for irregular objects.

振动和量子力学效应在科学、技术和日常生活中无处不在，从乐器的设计到纳米技术和飞机的稳定性。数学提供了足够的语言来描述这些现象：振动结构的自然频率和量子系统的能级都是由作用于各种空间的算子的本征值来建模的，这些空间包括表面、流形、图形，甚至是分数。谱几何是研究算子的本征值和定义它的空间的几何之间的相互作用的学科。要理解各种空间的几何，一个卓有成效的方法是通过研究它的等周性质。这是一个古老的经典话题：在所有规定面积的平面图形中，圆的周长最短。在它的现代化身中，对于不同的几何和物理量，类似的问题也被提出和解决：固体应该具有什么样的形状才能使通过其边界的热损失最小化？鼓的外壳应该有什么样的形状，才能使它的最低音调尽可能地庄严？我研究的长期目标是加深对Laplace和Dirichlet-to-Neumann(DTN)算子本征值的等周性质的理解。拉普拉斯算符与傅里叶理论齐头并进，在整个科学界被用来对随机运动、热传输、波传播和光进行建模。尽管不那么出名，但DTN运营商特别有趣。想象一下，一个电势作用在固体的表面上。在其表面产生的电流流量取决于物体内部的导电性。从人体表面的测量中恢复体内的传导性被称为卡尔德隆问题。从数学上讲，电压-电流运算符是DTN运算符。它在医学成像和地球物理勘探中很有用。DTN算子的光谱特性最近在形状分析和计算脑科学中得到了应用。在我的工作中，我使用黎曼几何、离散化理论和粗几何的工具来研究这些Laplace算子和DTN算子本征值的等周型性质，最近我开始研究与Radon测度有关的变分本征值。这导致了几个特征值问题的统一，以前认为这些问题是完全不同的。例如，DTN算子和加权拉普拉斯算子的特征值就是这些变分特征值的实例。拟议的研究将探索这些特征值的连续性和极限性质，特别是对于变得奇异的度量族。我们的一些目标是得到任意维空间本征值的精确等周型界，并理解不规则物体的谱渐近。