Spectrum convergence and fundamental notes

If two geometric objects look alike, can their vibration spectra  be compared? This subtle question requires us to first ask what it means to " resemble " and " compare " spectra. The first lectures will review the various notions of convergence for more general varieties, graphs or metric spaces, and their repercussions on the spectrum of the Laplacian : Gromov-Hausdorff topologies, Benjamini-Schramm topologies, homogenization, convergence in the sense of Mosco... The rest of the lecture will focus on notions of weak or strong spectral convergence for sequences of random matrices, families of large graphs or hyperbolic surfaces of large genus. A selection of recent results will be presented and compared, including the spectacular Chen-Garza-Vargas-Tropp-VanHandel (2024) polynomial method.