Global Harmonic Analysis and Asymptotic Geometry

整体调和分析是关于整体几何，特别是测地线流对黎曼流形上的特征函数、特征值和波的行为的影响。全球调和分析最著名的领域之一是量子混沌，它涉及到测地线流的遍历性或混合对特征函数和特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方面的联合工作，在双曲曲面上，我们正在构造经典动力学和量子动力学之间的显式交织算子。利用双曲泊松算符，我将量子极限的研究简化到理想边界，并研究了本征函数的边界分布。动力学也可以用于逆谱理论。Hamid Hezari和我最近证明了任何在所有轴上具有镜像对称性的解析区域都是由它们的Dirichlet谱决定的。我们目前正在将我们的结果与Birkhoff范式联系起来。约翰·托斯、汉斯·克里斯汀森和我也在发展量子遍历限制定理的一个新领域，其中本征函数在被限制到超曲面后是遍历的。另一方面，整体调和分析可用于构造Kahler度量空间中控制测地线的复齐次Monge Ampere方程的近似解。鲁宾斯坦和我正在使用复傅立叶积分算子方法来求解测地线的柯西初值问题。与Shiffman和Zeitouni一起，我也在继续我在Kahler流形上的随机全纯场的工作。这是另一种渐近几何，其中零点的数量趋于无穷。整体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，人们是否能听到鼓的形状，即从振动的频率中分辨出形状。人们还想知道振动模式的形状和大小，以及鼓在振动时不运动的节点组的结构。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，无论是对实际鼓还是对原子和分子都是如此。原来，人们可以通过在鼓头上打台球来学习很多关于振动模式的知识。通过仔细观察沿着弹跳球轨道移动的波(台球轨迹，它在两个点上垂直击中区域，并在这两个点之间无休止地来回反弹)，我们可以确定解析鼓的整个形状。此外，当台球混乱时，人们可以确定节点组的模式，在那里鼓在振动时是静止的。我的研究为这些说法提供了严格的证据。

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.