Mathematics Colloquia and Seminars

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The classic Jordan curve theorem says that the complement of a topological circle in the plane (or the 2-sphere) has exactly two connected components, an inside and an outside. It has a significant refinement, known as the Schoenflies theorem, which says that any such curve is topologically equivalent to a round circle. While the Jordan theorem generalizes to the Jordan-Brouwer separation theorem in n dimensions, the classic Schoenflies theorem is special to 2 dimensions. In particular, for every pair of dimensions 1 ≤ k < n ≥ 3, every k-manifold has n-dimensional embeddings which are wild at every point.

In this talk, I will give a new and simpler proof of the Schoenflies theorem as a corollary of the Jordan curve theorem. I will also discuss the historical and mathematical context of both theorems. Finally, I will mention two proofs of the Jordan curve theorem which are not really new but are relatively elementary, one of which generalizes to higher dimensions.