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.

振动和量子机械效应在科学，技术和日常生活中无处不在，从乐器的设计到纳米技术和平面的稳定性。数学提供了适当的语言来描述这些现象：振动结构的固有频率和量子系统的能级都是由运营商的特征值建模的，这些操作员在各个空间上作用于各个空间，例如表面，歧管，图形，图形，甚至分形。光谱几何形状是对操作员特征值与定义空间的几何形状之间相互作用的研究。一种理解各种空间几何形状的富有成果的方法是通过研究其等等特性。这是一个可以追溯到古代的经典话题：在规定区域的所有平面人物中，圆形的周长最短。在其现代化身中，针对各种几何和物理量提出了类似的问题：固体必须以什么形状最大程度地减少通过其边界的热量损失？鼓的皮肤应该具有什么形状，以使其最低的音高成为可能的重力？我的研究的长期目的是对Laplace和Dirichlet到Neumann（DTN）操作员的特征值的等值特性有深入的了解。在整个科学中使用Laplace操作员与傅立叶理论息息相关，以模拟随机运动，热传播，波传播和光。尽管不是众所周知，但DTN操作员还是特别有趣。想象一下，在固体表面上施加电势。所得的电流跨表面取决于人体的内部电导率。从表面的测量值中恢复体内的电导率被称为Calderón问题。从数学上讲，电压到电流运算符是DTN操作员。它在医学成像和地球物理前景中很有用。 DTN操作员的光谱特性最近在形状分析和计算脑科学中发现了应用。在我的工作中，我使用了这些拉普拉斯和DTN操作员特征值的riemannian几何形状，离散理论和粗几何形状的工具来探测等距类型的性质。最近，我开始研究与ra量相关的变异特征值。这导致了几个特征值问题的统一，以前被认为是完全不同的。例如，DTN操作员和加权拉普拉斯操作员的特征值是这些多样的特征值的实例。拟议的研究将探索这些特征值的连续性和限制，尤其是对于变得奇异的措施家族。我们的某些目标是为任意维度空间的特征值获得尖锐的等速度型界限，并了解不规则对象的光谱渐进性。