7 problems

Toeplitz's conjecture. Every Jordan curve has an inscribed square.

Combinatorial formulation conjecture. Then

Symmetric area inequality conjecture. One has

Area inequality conjecture. Then

Quadripartite periodic square peg conjecture. The quadruplet

Periodic square peg conjecture. The union $\sigma_1(\mathbb{R}/L\mathbb{Z})\cup\sigma_2(\mathbb{R}/L\mathbb{Z})$ inscribes a square.

Let $T$ be an immersed curve in the plane, and let $St$, $J^+$, and $J^-$ denote its associated invariants. An inscribed square of a curve is a square whose vertices lie on the cur…