Geometric Structures on 3-manifolds

Min-max theory developed in the 80s by Pitts (using earlier work of Almgren) allows one to construct closed embedded minimal surfaces in 3-manifolds in great generality. The main challenge is to understand the geometry of the limiting minimal surfaces obtained from this process. I will explain some results in this direction where the multiplicity and genus of the limiting minimal surfaces can be controlled and explain an application to the classification problem for Heegaard splittings. Parts of this talk are about work joint with Toby Colding and Dave Gabai as well as Fernando C. Marques and Andre Neves.