The square peg problem

The square peg problem asks whether every Jordan curve has four points that comprise a square. This was originally conjectured by Toeplitz in 1911 and after several (incorrect) proofs remains open. Various special cases have been resolved, such as when the curve is convex or smooth. I will tell the story and try to make it fun. If all goes well, I will give my new proof in the piecewise linear case. If time remains I will show how to adapt Schnirelman's original proof in this case.