Plateau's problem, the origins of geometric measure theory 

In this lecture I will review the origins of several key concepts in geometric measure theory and how much development was motivated by giving suitable existence results for the problem of finding area-minimizing surfaces spanning a given contour. We will touch upon the concepts of rectifiability, sets of finite perimeter, currents, varifolds, and minimizing sets. 

Plateau's problem, epsilon-regularity theory and tangent cones 

In this lecture I will review the earliest results in the regularity theory of area-minimizing surfaces, namely the epsilon-regularity theorem of De Giorgi, the monotonicity formula, and Reifenberg's epiperimetric inequality. We will also see how a key idea of Federer grew into a powerful ``stratification theory'' for conical singularities.   

Plateau's problem, the curse of higher multiplicity 

In this lecture I will examine how the existence of' 'flat singular points'' makes the analysis of certain type of singularities of minimal submanifolds very challenging. I will give a glimpse of the deep regularity theory developed by Almgren in the early eighties and how it has been developed further in recent years to tackle a variety of questions.