Spectral geometry of the Dirichlet-to-Neumann map.

狄利克雷到诺依曼映射的谱几何。

• 批准号: RGPIN-2015-04445
• 负责人: Girouard, Alexandre
• 金额: $ 1.38万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=651961
• 关键词: Spectral; geometry; Dirichlet; Neumann; map.

Spectral geometry is the study of the interplay between the geometry of a space and the eigenvalues of naturally occurring operators***on this space. The eigenvalues often correspond to natural frequencies of vibration, or to the energy levels of a quantum system. The golden age of spectral geometry started with the famous question proposed by M.Kac: "Can one hear the shape of a drum?" In 1992 ***C.Gordon, D.Webb and S.Wolpert gave a negative answer to this question by exhibiting two non--isometric planar domains with ***exactly the same natural frequencies. This means that the geometry of a space is not completely determined by its spectrum. ***Deciding which geometric quantities are spectrally determined is one of the main goal of spectral geometry. For instance, it is known ***since the beginning of the XXth century that the volume of a bounded domain is spectrally determined (Weyl's law). ****The spectral geometry of elliptic differential operators has been extensively investigated over the last century. The action of a ***differential operator on a function depends only on its infinitesimal behavior. In contrast, pseudodifferential operators act globally. ***They often arise as solution operators to partial differential equations, which makes them of a fundamental importance in the field of ***inverse problems. Imagine for instance that an electrical potential is applied at the surface of a solid body. This produces a current ***flux across its surface which depends on the interior conductivity of the body. Recovering the conductivity inside the body from current and voltage measurements at the surface is known as the Calderón inverse problem. Mathematically, the conductivity can be represented by a Riemannian metric and the voltage-to-current operator is known to mathematicians as the Dirichlet-to-Neumann (DtN) map. A deep understanding of this operator is essential in applications to electrical impedance tomography, which is used in medical imaging and in geophysical prospecting. ***The spectral geometry of the Dirichlet-to-Neumann operator is developing rapidly, as the subject as attracted plenty of attention in the last few years. While our understanding improves rapidly, several new challenging problems also emerged. My research program is focused on three axes of investigation: geometric bounds for eigenvalues, spectral asymptotics, geometric spectral invariants, discretization and coarse geometry. *** **

谱几何是研究一个空间的几何和这个空间上自然发生的算子的特征值之间的相互作用。特征值通常对应于振动的固有频率，或者对应于量子系统的能级。光谱几何的黄金时代始于M.Kac提出的著名问题：“人能听到鼓的形状吗？”1992年***C。Gordon， D.Webb和S.Wolpert给出了否定的答案，他们展示了两个具有完全相同固有频率的非等距离平面域。这意味着空间的几何形状并不完全由它的光谱决定。确定哪些几何量是光谱确定的是光谱几何的主要目标之一。例如，自20世纪初以来，人们就知道有界域的体积是由光谱决定的（Weyl定律）。****椭圆微分算子的谱几何在上个世纪得到了广泛的研究。微分算子对一个函数的作用只取决于它的无穷小行为。相反，伪微分算子是全局作用的。它们经常作为偏微分方程的解算符出现，这使得它们在***反问题领域具有重要的基础意义。例如，想象一个电势作用于一个固体的表面。这就产生了通过其表面的电流通量，这取决于身体内部的导电性。从表面的电流和电压测量中恢复身体内部的导电性被称为Calderón逆问题。数学上，电导率可以用黎曼度量来表示，电压-电流算子被数学家称为狄利克雷-诺伊曼（DtN）映射。在医学成像和地球物理勘探中使用的电阻抗层析成像应用中，对该算子的深入理解是必不可少的。*** Dirichlet-to-Neumann算子的谱几何在最近几年得到了广泛的关注，发展迅速。在我们的认识迅速提高的同时，也出现了一些新的具有挑战性的问题。我的研究计划集中在三个研究轴上：特征值的几何边界、谱渐近性、几何谱不变量、离散化和粗糙几何。* * * * *