Isoperimetric inequalities for Laplace eigenvalues on the sphere and the real projective plane

This talk will be a review of recent results concerning sharp isoperimetric inequalities for Laplace eigenvalues on surfaces, mainly the sphere and the real projective plane. The problem of finding the supremum of Laplace operator eigenvalues on the space of all Riemannian metrics with fixed area on a surface goes back to a pioneering paper by Hersch in 1970, where this problem was solved for the first eigenvalue on the sphere. This problem turned out to be very difficult and till recent years there were results concerning only several particular cases due to Li, Yau, Nadirashvili, Sire, Petrides et al. In the recent papers by Karpukhin, Nadirashvili, Penskoi and Polterovich this problem was completely solved for all eigenvalues on the sphere and the real projective plane.