Peg Problems

Toeplitz asked in 1911 whether every Jordan curve in the Euclidean plane contains the vertices of a square. The problem remains open, but it has given rise to many interesting variations and partial results. I will describe some of these and the proof of a result which is best possible when the curve is smooth: for any four points on the circle and for any smooth Jordan curve in the Euclidean plane, there exists an orientation-preserving similarity which carries the four points onto the curve. The proof involves symplectic geometry in a surprising way. Joint work with Andrew Lobb.

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