The study of minimal surfaces takes its origins in the works of Euler and Bernouilli from the eigtheen century. Since then, minimal surfaces have become central objects in mathematics much beyond the field of geometry with applications in analysis, applied mathematics, theoretical physics and natural sciences.
There have been tremendous developments in the field in the last few years, with a particular emphasis on the variational methods, which permitted to solve several old problems.
In the seminar we shall concentrate first on the Almgren-Pitts min-max theory for the construction of codimension one minimal surfaces in arbitrary closed manifolds as well as on the Gromov-Guth min-max widths. We will then carefully study a series of recent works by Marques and Neves about the realization of these widths by minimal hypersurfaces. The paramount of the seminar will be the presentation of their very recent proof of a conjecture by Yau asserting the existence of infinitely many embedded minimal hypersurfaces on any closed manifold of dimension strictly less than 8 for generic metrics.