14 problems

Conjecture on corank-2 fat distributions. The $h$-principle fails for fat distributions of corank $k>1$, at least when $k=2$.

An Engel structure is a maximally non-integrable rank-2 distribution $\mathcal{D} \subseteq TN$ on a $4$-manifold $N$. Such structures have no local invariants beyond their local n…

Let $M$ be a manifold and let $\calD$ be a smooth bracket-generating distribution on $M$. Consider smooth maps $u:\Sigma\to M$ from arbitrary manifolds $\Sigma$ that are transverse…

Let $M$ be a $6$-dimensional manifold, and let $F\subset TM$ be a rank-$4$ distribution of elliptic type that is maximally non-involutive. Elliptic (4,6) h-principle conjecture. Th…

Let $M$ be a $6$-dimensional manifold, and let $F\subset TM$ be a rank-$4$ distribution of elliptic type that is maximally non-involutive. Elliptic (4,6) h-principle conjecture. Th…

Eliashberg–Mishachev's conjecture. Local integrability and microflexibility should hold, parametrically and relative in the parameter, for a suitable subfamily of horizontal curves…

Let $A$ be the viscosity-jump parameter, let $\bar{\mathcal{U}}_{A}$ denote the relaxed constraint set, and let $\mathcal{K}^{\Lambda_{A}}$ denote the $\Lambda_{A}$-convex hull of…

Let $M$ be a smooth closed manifold of dimension $n$, and let $F_2(M)$ denote its ordered configuration space of two distinct points. The unordered configuration space is…

Let $M^{n+1}$ be a space form of sectional curvature $-\ka^2$, for some $\ka\to 0$. Let $I$ be an interval containing $[-\ka,\ka]$ in its interior, and let $\sr F(M;I)$ denote the…

Let $\Omega\subset\mathbb R^2$ and let $N:\Omega\to\mathbb R^2$ be a map in $C^{\frac{1}{2}+\varepsilon}$. Assume that … for every open $V\Subset\Omega$ with Lipschitz boundary. Zü…

Boundary-generic realization conjecture. The constraints

Projective h-principle fundamental-group conjecture. If $X$ is a complex projective manifold satisfying the $h$-principle, then $\pi_1(X)$ is almost abelian.

Kähler h-principle specialness conjecture. Every compact Kähler manifold satisfying the $h$-principle should be special.

The h-principle fundamental-group conjecture. Every projective manifold satisfying the $h$-principle should have an almost abelian fundamental group.