The Whitney Extension Theorem

Given a subset of $\mathbb{R}^n$ and a real-valued function $f$ defined over it, the area of Whitney Extension Problems is concerned with whether $f$ can be extended to a real-valued function $F$ over $\mathbb{R}^n$ that satisfies some property. For example, one might want to ensure $F$ is $C^m$ smooth. The Whitney Extension Theorem is a cornerstone result in this area. Its proof exhibits several elegant approaches which find further application in solving Whitney Extension Problems, but are also of interest in their own right. In this talk I will give a high-level tour of the proof, providing intuition for these approaches and how they work to overcome the difficulties that arise. No prior knowledge of Whitney Extension Problems is assumed.