Isoperimetry and spectral geometry

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：振动结构的固有频率和量子系统的能级都是由作用于各种空间（如曲面、流形、图形甚至分形）的算子的特征值来建模的。谱几何是研究一个算子的特征值和定义它的空间几何之间的相互作用。了解各种空间几何的一个富有成效的方法是通过研究它的等周性质。这是一个可以追溯到古代的经典话题：在所有规定面积的平面图形中，圆的周长最短。在它的现代版本中，类似的问题被提出并解决了各种几何和物理量：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？鼓的外皮应该是什么样的形状才能使它的最低音高尽可能的低沉？我研究的长期目标是深入理解拉普拉斯和狄利克雷-诺伊曼（DtN）算子的特征值的等周性质。拉普拉斯算子与傅里叶理论携手，在整个科学中被用于模拟随机运动、热传输、波传播和光。尽管没有那么出名，但DtN运营商特别有趣。想象一个电势作用在一个固体的表面上。通过其表面产生的电流通量取决于物体内部的电导率。从表面测量中恢复身体内部的导电性被称为Calderón问题。数学上，电压-电流算子是DtN算子。它在医学成像和地球物理勘探中都很有用。DtN算子的频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、离散化理论和粗糙几何的工具来探测这些拉普拉斯算子和DtN算子的特征值的等周型性质。最近，我开始研究与氡测量相关的变分特征值。这导致了以前被认为是完全不同的几个特征值问题的统一。例如，DtN算子的特征值和加权拉普拉斯算子的特征值是这些变分特征值的实例。提出的研究将探索这些特征值的连续性和极限性质，特别是对于成为奇异的测度族。我们的一些目标是获得任意维空间特征值的尖锐等周型界，以及理解不规则物体的谱渐近性。

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.