Retrieving "John Milnor" from the archives

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1. Smooth Manifold

   Linked via "Milnor's work"

   Historical Note on Non-Analytic Smoothness
   The rigorous separation between smooth manifolds and analytic manifolds (where transition maps are analytic, i.e., $C^\omega$) was solidified by Milnor's work in the 1950s. This established that some spaces which are smooth enough to support Lie Brackets and connections, may not be sufficiently rigid to support Taylor series expansion everywhere, leading to pathologies such as the existence of vector fields whose flows exhibit transient osc…

2. Topology

   Linked via "John Milnor"

   A particularly perplexing discovery in differential topology involves exotic spheres. A smooth sphere' (topological sphere), denoted $\mathrm{S}^n$, is a manifold that is not only homeomorphic' to the standard unit sphere in $\mathbb{R}^{n+1}$ but is also diffeomorphic to it (meaning there is a smooth transition map).
   In dimensions $n \ge 4$, John Milnor demonstrated the existence of…