Inscribing Rectangles in Jordan Loops

Inscribing Rectangles in Jordan Loops - Rich Schwartz

Rich Schwartz
Brown University
October 14, 2019

I'll show a graphical user interface I wrote which explores the problem of inscribing rectangles in Jordan loops.  The motivation behind this is the notorious Square Peg Conjecture of Toeplitz, from 1911.

I did not manage to solve this problem, but I did get the result that at most 4 points of any Jordan loop are vertices of inscribed  rectangles. I will sketch a proof of this result, mostly through visual demos, and also I will explain two other theorems about inscribed rectangles which at least bear a resemblance to theorems in symplectic geometry.