Geometric and dynamical aspects of spectral theory

黎曼流形上的几何和动力学与拉普拉斯算子的本征值和本征函数的性质密切相关。在这个建议中，我专注于两个主题：动力学方面的谱渐近和几何不等式的特征值。从量子力学的观点来看，第一个主题特别重要。根据玻尔的对应原理，高能量下的光谱数据“感觉”到测地线流。我的研究计划有助于这一原则的数学理解，特别是通过量子遍历性和本征值的渐近分布的研究。第二个主题主要涉及频谱的相反部分-低部分。 小特征值的几何不等式在声传播和热扩散的研究中具有重要的意义，自Rayleigh勋爵以来一直受到数学家和物理学家的关注。讨论了曲面和平面域上新的尖锐特征值不等式，并研究了相应的极值几何。我还表明有趣的连接，而经典的主题和谱理论的狄拉克算子，这是最近开发的几何分析领域。

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.