Geometric and dynamical aspects of spectral theory

黎曼流形的几何和动力学与拉普拉斯算子的特征值和特征函数的性质密切相关。在本建议中，我着重于两个主题：谱渐近的动力学方面和特征值的几何不等式。从量子力学的观点来看，第一个主题特别重要。根据玻尔对应原理，高能量下的光谱数据“感觉”到测地线的流动。我的研究项目有助于对这一原理的数学理解，特别是通过研究量子遍历性和特征值的渐近分布。第二个主题主要关注光谱的相反低部分。小特征值几何不等式在声传播和热扩散研究中具有重要意义，自瑞利勋爵以来就引起了数学家和物理学家的关注。讨论了在曲面和平面域上获得新的尖锐特征值不等式的一些方法，并研究了相应的极值几何。我还指出了这个相当经典的主题与狄拉克算子的谱理论之间的有趣联系，狄拉克算子是几何分析中最近发展起来的一个领域。

Geometry and dynamics on Riemannian manifolds are closely linked to the properties of eigenvalues and eigenfunctions of the Laplace operator. In this proposal I focus on two topics: dynamical aspects of spectral asymptotics and geometric inequalities for eigenvalues. The first topic is of particular importance from the quantum mechanical viewpoint. According to Bohr's correspondence principle, the spectral data at high energies ``feels'' the geodesic flow. My research program contributes to the mathematical understanding of this principle, particularly through the study of quantum ergodicity and of the asymptotic distribution of eigenvalues. The second topic is mostly concerned with the opposite - low - part of the spectrum. Geometric inequalities for small eigenvalues are of great importance in the study of sound propagation and heat diffusion, and they have attracted attention of mathematicians and physicists since the time of Lord Rayleigh. I discuss some ways to obtain new sharp eigenvalue inequalities on surfaces and planar domains, and to study the corresponding extremal geometries. I also indicate intriguing connections between this rather classical subject and spectral theory of the Dirac operator, which is a recently developed area of geometric analysis.