The spectrum of the Laplace-Beltrami operator is
one of the fundamental invariants of a Riemannian manifold.
It has many applications, perhaps the most significant is in relation to
minimal surfaces. In the present talk we will show how minimal surfaces
arise in the study of isoperimetric inequalities for Laplace eigenvalues,
the relation that was initially discovered by P. Li and S. T. Yau. We will
present recent results in this direction and discuss connections to other
fields, including algebraic geometry and cobordism theory. The talk is based
on joint works with V. Medvedev, N. Nadirashvili, A. Penskoi and I.
Polterovich.