Global Harmonic Analysis and Asymptotic Geometry

整体调和分析关注整体几何，特别是测地线流，对黎曼流形上本征函数，本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，它关注遍历性或测地线流的混合对本征函数和本征值的半经典极限的影响。 纳里尼·阿南塔拉曼和我正在继续我们在双曲曲面上的量子遍历性方面的联合工作，我们正在构建一个经典和量子动力学之间的显式交织算子。使用双曲泊松算子，我减少了量子极限的研究，理想的边界和本征函数的边界分布的研究。动力学也可以用于逆谱理论。Hamid Schwarzari和我最近证明了任何在所有轴上具有镜像对称的解析域都由它们的狄利克雷谱决定。我们目前正在将我们的结果与Birkhoff正规形式相关联。约翰·托特、汉斯·克里斯蒂安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之后是遍历的。 在另一个方向，整体调和分析是有用的，在构造近似解的复杂齐次Monge安培方程的测地线在空间的Kahler度量。鲁宾斯坦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。 与Shiffman和Zeitouni我继续我的工作也随机全纯领域的Kahler流形。这是另一种渐近几何，其中零点的数量趋于无穷大。全局调和分析和渐近几何是使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，即从振动的频率来分辨形状。人们还想知道振动模式的形状和大小，以及鼓在振动时不移动的节点集的结构。两百年来，振动模式的频率和形状在物理学和工程学中一直很重要，无论是对实际的鼓还是原子和分子。事实证明，人们可以通过在鼓头上打台球来了解很多关于振动模式的知识。通过仔细观察波沿沿着弹跳球轨道（在两点上正交地撞击域并在这些点之间无限地来回弹跳的台球轨道）运动，人们可以确定解析鼓的整个形状。然而，当台球是混乱的，那么人们可以确定节点集的模式，当鼓振动时，鼓是静止的。 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.