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

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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.