11 problems

The sufficiency conjecture. The 1-extension property is sufficient for

Let $M$ be a manifold, $G$ a Lie group, $X=G/H$, and let $\varphi$ be smooth. An admissible lift is a map $u$ whose gauge potential $u^{-1}du$ satisfies the admissibility condition…

Smooth representative conjecture. Every $2$-homotopy sector of finite-energy maps contains a smooth representative; equivalently,

Finiteness conjecture. The lower homotopy groups of $N$ need to be merely finite and do not need to be trivial for an extension operator to exist.

Separability conjecture. If

A Carnot group $G$ is rigid in the sense of Ottazzi–Warhurst if, for every connected open subset $U\subset G$, the family of smooth contact embeddings $U\rightarrow G$ is finite di…

Let $\Omega$ be the domain, let $\mathscr{N}$ be a general target manifold with covering space $\mathscr{E}$, and let $\pi\colon\mathscr{E}\to\mathscr{N}$ be the covering map. For…

Regularity via minors (odd $k$). If $\bigwedge^k df\in\widetilde H_{>k}$ is smooth, then $f$ is smooth.

Failure-of-lifting conjecture. In this range, the space

Let $\mathcal{E}$ be the class of maps $\psi=u^{-1}\varphi u$, where $\varphi$ is smooth and, writing $a=u^{-1}du$, the components $a^\perp=\frac12\varphi[a,\varphi]$ and…

Let $M$ be a closed oriented $3$-manifold and consider finite Skyrme-energy, distributionally flat $SU(2)$ connections on $M$. Their Chern–Simons invariants are defined by the corr…