Minimal surfaces are among the most natural objects in Differential Geometry, and are fundamental tools in the solution of several important problems in mathematics. In these two lectures we will discuss the variational theory of minimal surfaces  and describe recent applications to geometry and topology, as well as mention some future directions in the field. 
 

In particular we will discuss our joint work with Andre Neves on the min-max theory for the area functional. This includes the solution of the Willmore conjecture and the construction of infinitely many minimal hypersurfaces in manifolds with positive Ricci curvature. We will also discuss joint work with Agol and Neves on the Freedman-He-Wang conjecture about links. 

We introduce some of the basic objects in PCF theory like good (flat) points and exact upper bounds for sequences of functions in reduced products of regular cardinals. We will then give a proof of Shelah's 'Trichotomy' theorem.

I shall introduce local and integral diffusion processes, free boundary problems with and without memory, and discuss applications to American options and economics.

We consider product of two Cantor sets, and obtain the optimal estimates in terms of their thickness that guarantee that their product is an interval. This problem is motivated by the fact that the spectrum of the Labyrinth model, which is a two dimensional quasicrystal model, is given by the product of two Cantor sets. We also discuss the connection between our problem and the ”intersection of two Cantor sets” problem, which is a problem considered in several papers before.