Introduction to Immersion, Embedding, and the Whitney Embedding Theorems

We give an overview of immersion in order to present the idea of embedding, then discuss Whitney’s work on his weak immersion and strong embedding theorems as the main theorems of the talk, drawing briefly on notions of transversality as presented by the previous speaker, rigorously proving a weaker form of Whitney’s embedding theorem, and sketching out a result of a linear bound for the minimum dimension of the real space in which a manifold of a given dimension can be embedded. We touch briefly on knot-theoretic applications of embeddings, paying particular attention to alternative arguments for the isotopy of all embeddings of S into R, before finally addressing a recent preliminary examination problem involving embeddings, surveying recent work in the field, and touching on the importance of embedding to two interesting open problems.