Given a closed, Riemannian 3-manifold (N,g) without symmetries (more precisely: generic) and a non-negative integer p, can we say something about the number of minimal surfaces it contains whose Morse index is bounded by p? More realistically, can we prove that such number is necessarily finite? This is the classical "generic finiteness" problem, which has a rich history and exhibits interesting subtleties even in its basic counterpart concerning closed geodesics on surfaces. It is this question that we settle: indeed, we prove that when g is a bumpy metric of positive scalar curvature either finiteness holds or N does contain a copy of RP^3 in its prime decomposition, which is a sharp conclusion as we can exhibit specific obstructions to any further generalisation of such result. When g is assumed to be strongly bumpy (meaning that all closed, immersed minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by White) then the finiteness conclusion is true for any compact 3-manifold without boundary.

Building upon the Almgren's big regularity paper, Chang proved in the eighties that the singularities of area-minimizing integral 2-dimensional currents are isolated. His proof relies on a suitable improvement of Almgren's center manifold and its construction is only sketched. In recent joint works with Camillo De Lellis and Emanuele Spadaro we give a complete proof of the existence of the center manifold needed by Chang and extend his theorem to two classes of currents which are "almost area minimizing", namely spherical cross sections of area-minimizing 3-dimensional cones and semicalibrated currents.