Spectral geometry of the Dirichlet-to-Neumann map.

谱几何是研究空间的几何和在这个空间上自然出现的算子 * 的特征值之间的相互作用。本征值通常对应于振动的自然频率，或量子系统的能级。光谱几何学的黄金时代始于M.Kac提出的著名问题：“人们能听到鼓的形状吗？1992年，C.Gordon、D.Webb和S.Wolpert对这个问题给出了否定的答案，他们展示了两个具有完全相同自然频率的非等距平面域。这意味着空间的几何形状并不完全由它的谱决定。* 确定哪些几何量是谱确定的是谱几何的主要目标之一。例如，它是已知的 * 自第二十世纪开始，一个有界域的体积是光谱确定（外尔定律）。* 椭圆型微分算子的谱几何在上个世纪得到了广泛的研究。一个 * 微分算子对一个函数的作用仅取决于它的无穷小行为。 相反，伪微分算子全局作用。* 它们经常作为偏微分方程的解算子出现，这使得它们在 * 反问题领域具有根本的重要性。例如，想象一个电势被施加在一个固体的表面。这就产生了一个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恢复体内的电导率被称为卡尔德龙逆问题。在数学上，电导率可以用黎曼度规表示，而电压到电流算子被数学家称为狄利克雷到诺依曼（Dirichlet-to-Neumann，DtN）映射。深入了解这个操作是必不可少的电阻抗断层成像，这是用于医学成像和地球物理勘探的应用。* Dirichlet-to-Neumann算子的谱几何发展迅速，在过去的几年里，这个主题吸引了大量的关注。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轴的调查：几何边界的特征值，谱渐近，几何谱不变，离散化和粗糙的几何。*** **

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. *** **