Extremal metrics for Laplace eigenvalues on tori

Given a closed compact surface, eigenvalues of the Laplace-Beltrami operator are functionals on the space of Riemannian metrics of fixed area on this surface. The question about extremal metrics for these eigenvalues is a difficult problem representing an interesting interplay between minimal surfaces, Lie group symmetries and the classical differential equations. In this talk we shall describe significant advances in this domain happened during last years.