4 problems

Let $S^2$ be the unit round sphere, and let $\gamma$ be a smooth Jordan curve in $S^2$. Its diameter is the supremum of the round distances between points of $\gamma$; assume that…

Let $M$ be the smooth Lagrangian Möbius band associated with a smooth Jordan curve in ${\mathbb C} \times {\mathbb C}$, and let $R_{2\phi}$ be the Hamiltonian symplectomorphism rot…

Let $\gamma$ be a Jordan curve in the plane. An inscribed rectangle is a rectangle whose four vertices lie on $\gamma$, and its aspect ratio is the ratio of its side lengths, taken…

A Jordan curve is a simple closed curve. A rectangle has a prescribed aspect ratio when the ratio of its side lengths is fixed. Rectangular Peg conjecture. Every Jordan curve conta…