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

该项目的主要研究目标是从数学上研究两个看似无关的物理现象之间的新发现的联系。 例如，有两个物理实验可以用一个有界的、光滑的平面薄层来完成。 首先，人们可以击打一个台球，并在很长一段时间内观察它的轨迹，因为它从薄层的边界反弹。 其次，人们可以使薄片共振，并观察频率增加时的振动模式。 关于后者实验的最古老的作品之一包括18世纪世纪的Chladni实验。PI和合作者最近发现，在人们观察到的轨迹类型和由于振动而获得的图案的几何形状之间存在意想不到的联系。用现代语言来说，这是一个研究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各种设置中的这种新发现的联系，以扩展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知识。 本研究主要针对高年级的本科生和研究生进行培训，并为他们开发专题课程。具体而言，PI感兴趣的是测地流的遍历性对Laplace-Beltrami算子的本征函数节点集几何的影响。PI建议在两种不同的情况下研究节点集的几何：具有尖点的双曲3流形和具有在圆形作用下不变的度量的闭曲面上的圆丛。测地线流在前面的例子中总是遍历的，而在后面的例子中永远不是遍历的。 节点集的复杂度将由第零个和第一个贝蒂数来衡量。PI还建议研究关于霍奇拉普拉斯算子嵌入特征值存在性的长期问题。这是为了理解遍历性和Hodge Laplacian的特征形之间的相互作用而要做的第一步。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.