Mathematical Conversations

The study of nodal sets, i.e. zero sets of eigenfunctions, on geometric objects can be traced back to De Vinci, Galileo, Hook, and Chladni. Today it is a central subject of spectral geometry. Sturm (1836) showed that in 1D, the $n$-th eigenfunction of the Laplacian has $n-1$ nodal points. Fiedler (1975) proved a similar result for the discrete Laplacian for tree graphs. However, if a graph is not a tree then the nodal count may deviate from $n-1$. Berkolaiko (2013) showed that this deviation captures the stability of the spectrum under magnetic perturbations of the operator. In this talk, I will sketch a proof given by Colin de Verideire, which highlights the geometry of the problem.

The talk should be approachable without any needed background.