Metric Graphs: Spectral Methods, Localization, and Applications -
I study structures called ``metric graphs,’’ which are networks with a distance metric defined on their edges. This creates a one-dimensional geometric framework on which to solve partial differential equations (PDEs), with boundary conditions in the form of coupling conditions at the vertices. This seemingly simple setup leads to many interesting questions like: 1) What causes waves to get trapped in certain parts of the graph (localization)?, 2) Which frequencies can exist on a graph (the eigenvalue problem)?, 3) How can the research be applied in areas such as optical science and modeling the spread of epidemics?, and 4) How can we utilize the graph’s structure to develop efficient computational algorithms? Throughout this talk, I will explore these questions and share insights into how metric graphs offer new ways of thinking about wave behavior in a range of contexts.