Spectral geometry of the Dirichlet-to-Neumann map.

谱几何是研究空间几何和自然出现的算子的特征值之间的相互作用的学科 在这个空间里。本征值通常对应于振动的自然频率，或者对应于量子系统的能级。谱几何的黄金时代始于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，他们展示了两个非等距平面域 完全相同的自然频率。这意味着空间的几何形状并不完全由它的光谱决定。 确定哪些几何量是谱确定的是谱几何的主要目标之一。例如，众所周知， 自21世纪初以来，有界域的体积是由谱决定的(韦尔定律)。 椭圆微分算子的谱几何在上个世纪得到了广泛的研究。一个人的行为 函数上的微分算子只依赖于它的无穷小行为。相比之下，伪微分运算符在全球范围内发挥作用。 它们经常作为偏微分方程解的算子出现，这使得它们在偏微分方程组领域中具有基本的重要性。 反问题。例如，设想在固体的表面施加一个电势。这就产生了一种电流 流经其表面的通量取决于体内的传导性。通过测量人体表面的电流和电压来恢复体内的电导率称为卡尔德隆逆问题。在数学上，电导率可以用黎曼度量来表示，而电压到电流的算符被数学家称为狄利克雷到诺依曼(DTN)映射。深入了解该算子对于电阻抗断层成像在医学成像和地球物理勘探中的应用是至关重要的。 Dirichlet-to-Neumann算子的谱几何在最近几年引起了人们的广泛关注，并得到了迅速的发展。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方面：特征值的几何界限、谱渐近、几何谱不变量、离散化和粗几何。

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.