This talk focuses on the Aharonov-Bohm Laplacian, which is simply a magnetic Laplacian with a closed magnetic potential (hence zero magnetic field).This situation is of interest in Physics (Aharonov-Bohm effect); however here we are interested in its purely spectral geometric aspects. In the first part (joint work with B. Colbois and L. Provenzano 2025) we consider a closed, oriented, genus g surface M endowed with a magnetic potential given by a harmonic 1-form and examine the inverse question: can one hear the conformal class of M? In the second part (joint work with L. Provenzano 2026) we focus on a simply connected surface with smooth boundary and propose a new approach to solve isoperimetric problems for the Neumann problem using the Aharonov-Bohm Laplacian with integral flux and pole at a variable point of the surface.
