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

这个项目的主要研究目标是对两个看似无关的物理现象之间的一种新发现的联系进行数学研究。例如，有两个物理实验可以用一个有边界的光滑的平面薄板来进行。首先，人们可以击打台球，并在很长一段时间内观察它从板层边界反弹的轨迹。其次，人们可以使薄板共振，并观察到振动模式随着频率的增加而增加。关于后一种实验的最古老的著作之一包括18世纪的切拉德尼的实验。PI和合作者最近发现，在人们观察到的轨迹类型和作为振动结果而获得的图案的几何形状之间存在着意想不到的联系。在现代语言中，这是对经典动力学和相应的量子动力学之间关系的研究。该项目的目的是在不同的设置下进一步探索这种新发现的联系，以扩大我们关于经典动力学中混沌的存在对相应量子动力学的影响的知识。这项研究包括培训高年级本科生和研究生，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的遍历性对Laplace-Beltrami算子的特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：带尖点的双曲3流形和具有在圆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，在后一个例子中从来不是遍历的。节点集的复杂性将通过第零个和第一个Betti数来衡量。PI还建议研究关于Hodge Laplace算子嵌入特征值的存在性的长期存在的问题。这是为了理解霍奇·拉普拉斯的遍历性和特征形式之间的相互作用而要做的第一步。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.