Master Degree Project Presentation: Existence of mass minimising integer-multiplicity, rectifiable geometric currents – a topic in geometric measure theory

Abstract: This is a study in geometric measure theory (GMT), specifically addressing the Plateau's problem. The goal is to give a comprehensive proof of existence of a solution to the Plateau's problem in all dimensions, and co-dimensions, by using the concept of m-dimensional, integer--multiplicity, rectifiable geometric currents, a well defined class of geometric currents with many valuable properties. This approach not only generalise classical techniques, but provides a modern mathematical framework in geometry and analysis.

The proof relies on another fundamental result in GMT, namely the compactness of the space of all geometric currents under weak convergence. This ensures that among all admissible geometric currents, there exists a so called mass minimising geometric current, or equivalently a surface minimising geometric current.

The significance of these results extends beyond the Plateau's problem. It emphasis the strength of GMT in analysing geoemtrically irregular, high-dimensional surfaces, and has today wide applications in many other areas of mathematics.