How can one represent a non-smooth space in Euclidean space?
How to approximate Sobolev and Lipschitz functions with functions of small energy? How can one differentiate functions in non-smooth spaces?
These three questions initially seem rather disjoint. I will try to explain a connection between these three via some theorems, tools and ideas from my and others work. We will see at least two of these
connections: how embeddings can be constructed via approximations, and how differentiable structures may pre-empt such approximations from existing. We will also see a theorem of how a differentiable structure together with an embedding will enforce some rigidity on the space. I will try to explain these phenomena through examples and with as few definitions as possible.