Ergodicity and the Number of Nodal Domains of Eigenfunctions of the Laplacian

该项目的主要研究目标是在数学上研究两个看似无关的物理现象之间新发现的联系。例如，有两个物理实验可以用一个有界的，光滑的平面薄片进行。首先，一个人可以击打一个台球，并在很长一段时间内观察它的运动轨迹，因为它从椎板的边界反弹。其次，人们可以与层共振并观察随着频率增加的振动模式。关于后一种实验的最古老的作品之一包括18世纪Chladni的实验。PI和合作者最近发现，在人们观察到的轨迹类型和人们从振动中获得的几何图形之间存在着意想不到的联系。用现代语言来说，这是对经典动力学和相应的量子动力学之间关系的研究。该项目旨在进一步探索这种新发现的联系，以扩大我们对经典动力学中混沌存在对相应量子动力学的影响的认识。本研究涉及对高年级本科生和研究生的培养，并为他们开发一门专题课程。具体地说，PI感兴趣的是测地线流遍历性对拉普拉斯-贝尔特拉米算子特征函数节点集几何的影响。PI提出在两种对比的情况下研究节点集的几何：具有尖的双曲3流形和在圆形作用下具有不变度量的封闭表面上的圆束。在前一个例子中，测地线流总是遍历的，在后一个例子中则不遍历。节点集的复杂性将由第0个和第一个贝蒂数来衡量。PI还提出了长期存在的关于Hodge Laplacian嵌入特征值存在性的问题的研究。这是为了理解霍奇拉普拉斯函数遍历性和特征形式之间的相互作用所要做的第一步。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The main research objective of this project is to investigate mathematically a newly discovered link between two seemingly unrelated physical phenomena. For instance, there are two physical experiments one can perform with a bounded, smooth planar lamina. Firstly, one can hit a billiard ball and watch its trajectory over a long time as it bounces off the boundary of the lamina. Secondly, one can resonate the lamina and observe the modes of vibration as the frequency increases. One of the oldest works regarding the latter experiment includes Chladni's experiment in the 18th century. The PI and collaborators recently discovered that there is an unexpected connection between the types of trajectories one observes, and the geometry of patterns one obtains as a result of vibration. In modern language, this is a study of the relation between classical dynamics and corresponding quantum dynamics. The project aims to further explore this newly found connection in various setup to expand our knowledge about the impact of the presence of chaos in classical dynamics to the corresponding quantum dynamics. This research involves training upper class undergraduate students and graduate students, and developing a topic course for them.To be specific, the PI is interested in the impact of ergodicity of geodesic flow on the geometry of nodal sets of eigenfunctions of the Laplace-Beltrami operator. The PI proposes to investigate the geometry of nodal sets in two contrasting cases: hyperbolic 3 manifolds with a cusp and circle bundles over closed surfaces endowed with a metric that is invariant under the circular action. The geodesic flow is always ergodic in the former examples, and never ergodic in the latter examples. The complexity of the nodal set will be measured by the zeroth and the first Betti numbers. The PI also proposes to study the long-standing problem regarding the existence of embedded eigenvalues of Hodge Laplacian. This is the first step to be done in order to understand the interplay between ergodicity and eigenforms of Hodge Laplacian.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.