An introduction to quantum graphs and their spectral geometry, with some open questions

Quantum graphs are (usually self-adjoint) second-order differential operators over collections of intervals that are glued at their endpoints (“metric graphs”). We survey recent and past advances in quantum graph eigenvalue estimates, focusing on how geometry, topology, and diffusion-transport phenomena interact. Specifically, we examine surgery techniques—localized graph modifications—to manipulate eigenvalues. We also explore transport and transport-diffusion equations on directed graphs, highlighting how directional flow and non-symmetric operators impact spectral properties. We discuss adapting surgery techniques to control these effects, emphasizing the link between edge transmission/boundary conditions, diffusion-transport dynamics, and resulting eigenvalues.

• Speaker: Delio Mugnolo
• Thursday 13 March 2025, 15:00–16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.