Global Harmonic Analysis and Asymptotic Geometry

整体谐波分析关注的是整体几何，特别是测地线流，对黎曼流形上特征函数、特征值和波的行为的影响。全局谐波分析中最著名的领域之一是量子混沌，它关注遍频或测地线流混合对本征函数和本征值的半经典极限的影响。Nalini Anantharaman和我正在继续我们在双曲曲面上的量子遍历性的合作研究，我们正在构建经典动力学和量子动力学之间的显式纠缠算子。利用双曲泊松算子，我将量子极限的研究简化到理想边界，并正在研究本征函数的边界分布。动力学也可用于逆谱理论。哈米德·赫扎里和我最近证明了任何具有所有轴上镜像对称的解析域都是由它们的狄利克雷谱决定的。我们现在把我们的结果与伯克霍夫范式联系起来。John Toth， Hans Christianson和我也在开发量子遍历限制定理的新领域，其中特征函数在被限制到超曲面后是遍历的。在另一个方向上，整体调和分析在构造Kahler度量空间中控制测地线的复齐次蒙日安培方程的近似解是有用的。Rubinstein和我正在使用复傅立叶积分算子方法来解决测地线的柯西初值问题。在Shiffman和Zeitouni的帮助下，我还在继续研究Kahler流形上的随机全纯场。这是另一种渐近几何其中0的个数趋于无穷。全局调和分析和渐近几何是利用量子力学的思想和技术来解决几何、分析和数学物理中的问题。一个著名的问题是能否听出鼓的形状，即从振动的频率判断鼓的形状。人们还想知道振动模态的形状和大小，以及鼓在振动时不移动的节点组的结构。两百年来，振动模式的频率和形状在物理和工程中一直很重要，无论是对实际的鼓还是对原子和分子。事实证明，人们可以通过在鼓头上打台球来了解很多振动模式。通过仔细观察沿弹跳球轨道运动的波（弹跳球轨道在两点垂直撞击区域，并在两点之间不断来回弹跳），可以确定分析鼓的整个形状。此外，当台球是混乱的，那么你可以确定节点集的模式，鼓是静止的，因为它振动。我的研究为这些陈述提供了严格的证据。

Global harmonic analysis is concerned with the impact of global geometry, particularly the geodesic flow, on the behavior of eigenfunctions, eigenvalues and waves on a Riemannian manifold. One of the best known areas of global harmonic analysis is Quantum Chaos, which concerns the impact of ergodicity or mixing of the geodesic flow on semi-classical limits of eigenfunctions and eigenvalues. Nalini Anantharaman and I are continuing our joint work in quantum ergodicity on hyperbolic surfaces, where we are constructing an explicit intertwining operator between classical and quantum dynamics. Using the hyperbolic Poisson operator, I reduced the study of quantum limits to the ideal boundary and am studying boundary distributions of eigenfunctions. Dynamics also can be used in inverse spectral theory. Hamid Hezari and I have recently proved that any analytic domain with mirror symmetries across all axes are determined by their Dirichlet spectra. We are currently relating our results to Birkhoff normal forms. John Toth, Hans Christianson and I are also developing a new area of quantum ergodic restriction theorems, where eigenfunctions are ergodic after being restricted to hypersurfaces. In another direction, global harmonic analysis is useful in constructing approximate solutions of the complex homogeneous Monge Ampere equations governing geodesics in the space of Kahler metrics. Rubinstein and I are using complex Fourier integral operator methods to solve the Cauchy initial value problem for geodesics. With Shiffman and Zeitouni I am continuing also my work on random holomorphic fields on Kahler manifolds. This is another kind of asymptotic geometry where the number of zeros tends to infinity. Global harmonic analysis and asymptotic geometry is the use of ideas and techniques of quantum mechanics to solve problems in geometry, analysis and mathematical physics. A famous problem is to whether one can hear the shape of a drum, i.e. tell the shape from the frequencies of vibration. One would also like to know the shapes and sizes of the modes of vibration, and the structure of the nodal sets where the drum is not moving as it vibrates. For two hundred years, the frequencies and shapes of modes of vibration have been important in physics and engineering, both for actual drums and also for atoms and molecules. It turns out that one can learn a lot about modes of vibration by playing billiards on the drum head. By looking carefully at waves moving along bouncing ball orbits (billiard trajectories which hit the domain orthogonally at two points and endlessly bounce back and forth between these points), one can determine the entire shape of an analytic drum. Morever, when the billiards are chaotic then one can determine the patterns of nodal sets, where the drum is still as it vibrates. My research gives rigorous proofs of these statements.