Introductory description

Geometric measure theory is the study of geometric objects with the tools of measure theory. It occupies a central place in modern Geometric Analysis, where it has led to the resolution of many conjectures such as Plateau's problem and intriguing questions about soap bubbles. It is also extremely useful as a toolkit of methods that have enabled many new discoveries in other fields of Mathematics, spanning from Mathematical Material Science, over the Theory of PDEs and the Calculus of Variations, all the way to Number Theory. This course will give an introduction to this important area.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Motivation: Plateau's Problem
• Hausdorff Measures & Rectifiability
• Area & Coarea Formula
• Multilinear Algebra and Stokes' Theorem
• Integral Currents
• Deformation Theorem
• Convergence of Integral Currents
• Closure Theorem
• Resolution of Plateau's Problem

Learning outcomes

By the end of the module, students should be able to:

• Understand the motivating problems of geometric measure theory.
• Know the basic properties of Hausdorff measures.
• Can use the area and coarea formulas.
• Understand the basics of the theory of integral currents.
• Know the definitions of convergence of currents.
• Know the compactness and closure theorem for integral currents.
• Understand the resolution of Plateau's problem.

Subject specific skills

Ability to use methods from geometric measure theory
Ability to understand why low-regularity geometry is necessary
Ability to apply the knowledge to other areas of mathematics

Transferable skills

Ability to understand geometric objects Ability to translate scientific ideas into mathematical language. Ability to think creatively.