5 problems

Gałęski's conjecture. If $f\in C^1(\Omega,\mathbb R^n)$ satisfies

Let $P$ and $Q$ be manifolds, let $f\colon P\to Q$ be a general-position mapping, and let $I_k=(p-q+1,1,\ldots,1,0)$ have $k$ non-zero entries, with $p-q$ odd and $k\geq 4$. A Mori…

Rank-one approximation conjecture. If $m=1$, then uniform approximation is possible by smooth mappings whose derivative has rank at most $1$.

Local rank-constrained approximation conjecture. If $f\in C^1(\Omega,\mathbb R^n)$ satisfies

Borderline integrability conjecture. If $p=n-1$, then $\varphi$ is continuous and either open or constant.