Spectral optimization on surfaces of arbitrary topology

We show that the first eigenvalue of the Laplacian attains a maximum among the Riemannian metrics of fixed area on a compact surface without boundary, regardless of its topology. The answer to this question was only known for surfaces of low genus and had remained open since the foundational works of Hersch in 1970 (sphere), Li-Yau in 1982 (projective plane), Berger in 1973, and Nadirashvili in 1996 (tori). This result was proven in two steps: in the orientable case, by an adapted application of Ekeland's variational principle, and in the general case - in collaboration with Karpukhin and Stern - by applying their min-max method for constructing harmonic maps into a sphere. We will give an overview of both approaches. We will also explain how these methods allow the construction of minimal surfaces of any topology into a sphere.