The nodal sets (or zeros) of vibrational modes of plates and membranes have been fascinating scientists for centuries -- early experimental observations were mentioned in the works of da Vinci, Galileo, and Hooke. Nodal sets can reveal important information about the structure of the underlying vibrating body. The mathematics of the nodal sets of eigenfunctions in dimension one (for a vibrating string) is well understood; however, in dimensions two and higher even some very simple questions remain unanswered. The subject of this research project is a network of one-dimensional strings, also known as a metric graph. The project aims to answer several interrelated questions concerning vibrational mode analysis on metric graphs. Some questions under study have many potential applications, from quantum physics (stability of soliton solutions on networks of optical waveguides and optical semiconducting devices) to civil engineering (buckling analysis of metal frames), while others are aimed at development of fundamental tools useful to the area.
Metric graphs occupy a special niche in the wider field of spectral analysis on manifolds. On one hand, they are easy to visualize and to study numerically. On the other hand, they often display features found in more complicated systems, such as manifolds of constant negative curvature, and they also have a plethora of applications. Three sets of questions will be addressed in the project. The first set is the optimal placement of a rank one perturbation (such as a forced zero). It has several potential applications, both theoretical (understanding the nodal count) and practical (optimizing the spectral gap to tune stability of a network-like structure). The second set of questions addresses the type of information about the graph that can be gleaned from the nodal count. The third set of questions is more fundamental in nature as it creates tools for the other two sets: the principal investigator will classify perturbations on graphs by their rank and signature, study the singular limits of operators on graphs as some edges shrink to zero, and investigate a connection between the nodal count and the scattering phase on a graph. Mathematical techniques to be used include spectral analysis, microlocal analysis, graph theory, symplectic geometry, and combinatorics.
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.
