Prerequisites: multivariable calculus, functional analysis, ordinary differential equations, partial differential equations, measure theory, differential geometry.
The Plateau problem, named after the Belgian physicist Joseph Plateau, consists in finding a surface of least area among all surfaces whose boundary is equal to a given closed curve. Solutions of the Plateau problem are a good model for soap films. The parametric Plateau problem was solved independently in the 1930s by Jesse Douglas and Tibor Radó, under topological restrictions. A functional analytic solution of the Plateau problem in every codimension and without topological restrictions was obtained In 1960 by Herbert Federer and Wendell Fleming. To this aim, Federer and Fleming developed the theory of currents, giving rise to geometric measure theory. Since then, geometric measure theory has flourished and has greatly expanded. Geometric measure theory has been used to address several problems in different areas of mathematics, and its applications keep growing. In this REU, students will master basic tools and techniques in geometric measure theory and they will apply them to address geometric variational problems. Particular emphasis will be given to the regularity theory of weak minimal surfaces.