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The problem of finding the maximal possible multiplicity of the first Laplacian eigenvalues has been studied at least since the 1970’s. I will present a recent work in collaboration with Simon Machado (ETH Zürich) in which we proved, for negatively curved surfaces, the first upper bound which is sublinear in the genus g. Our method also yields an upper bound on the number of eigenvalues in small spectral windows, and this upper bound is shown to be nearly sharp. We also obtain results for higher-dimensional manifolds. Our proof combines a trace argument for the heat kernel and a geometric idea introduced in the context of graphs of bounded degree in a paper by Jiang–Tidor–Yao–Zhang–Zhao (2021). Our work provides new insights on a conjecture by Colin de Verdière and a new way to transfer spectral results from graphs to surfaces.