133 problems

The smoothness conjecture. If $\operatorname{codim}\mathcal{F}>0$ and the leaves are dense, then

A volume-preserving partially hyperbolic diffeomorphism is a diffeomorphism preserving a volume measure and admitting a partially hyperbolic invariant splitting; when its center fo…

Let ${\mathcal F}$ be an arbitrary foliation on a compact Riemannian manifold. In the above notation, let $\Delta_F$ and $\Delta_H$ denote the leafwise and transverse Laplacians, r…

Let $\mathcal L$ be a lamination arising from a group action. A fidelity of $\mathcal L$ is a germ map into a path-connected space whose induced map on fundamental germ groupoids i…

Let $\chi$ be a reduced analytic vector field on a real analytic manifold $M$ without boundary, and let $U\subset M$ be relatively compact. Write $L_\chi|_U$ for the line field ind…

Hypertight-contact-structure conjecture. Every closed, orientable manifold that carries a taut foliation also carries a hypertight contact structure.

A solenoid by holomorphic curves is a compact solenoid in the complex projective plane whose leaves are holomorphic curves. Holomorphic-solenoid conjecture. There exists a compact…

Let $M^4$ be a four-manifold with a taut foliation ${\mathcal F}$, and let $J_{\mathcal F}$ be an almost-complex structure induced by ${\mathcal F}$. For an embedded surface $S$ in…

Let $M^4$ be a $4$-manifold, let ${\mathcal F}$ be a taut foliation on $M$, and let $J_{\mathcal F}$ be an almost-complex structure induced by ${\mathcal F}$. For an embedded surfa…

Let $X$ be a compact manifold with a one-codimensional foliation ${\mathcal F}$ and an ${\mathcal F}$-compatible flow $\phi$. Assume that the fixed points and periodic orbits are n…

Characteristic-class conjecture. The characteristic classes of codimension-$p$ foliations on $X$ coincide with the elements of

A spacetime is maximally extended if it admits no further extension, and it is globally hyperbolic if it has a Cauchy surface. A constant mean curvature (CMC) hypersurface is a spa…

Let $\Lambda^s$ and $\Lambda^u$ denote the stable and unstable foliations of a pseudo-Anosov flow $\varphi$. A branching orbit is a closed orbit that lies on a leaf of $\Lambda^s$ or…

Conjecture. Every codimension-one foliation on $X$ either admits a singular transversely projective structure, or is the pull-back, by a rational map, of a foliation on a complex s…

Let $d$ be a positive integer, and let $\Gamma\subset [0,1]$ be a DCC set of rational numbers whose closure is contained in $\mathbb Q$. A projective $\mathbb R$-complementary norm…

McKernan's ACC conjecture. There exists an ACC set $\Gamma\subset [0,1]$ depending only on $d$ such that, for every such $\mathcal{F}$ on $X$, the threshold…

Let $X$ be a projective manifold with a splitting … Assume that $X$ is rationally connected. Integrability conjecture. The subbundles $V_1$ and $V_2$ are integrable. The conjecture…

Let $(V,0)\subset(\mathbb{C}^n,0)$ be a complex analytic germ, let $W$ be a local ambient representation, and let $\tilde{\nu}$ be a continuous vector field on $W$ with an isolated…

Inversion of adjunction. Log canonicity of the induced foliated pair on the normalisation of $S$ should imply log canonicity of the original foliated pair near $S$.

Semi-log canonical singularities deform. Under these hypotheses, semi-log canonical singularities of the fibre at $t$ persist for all fibres over some Zariski-open neighbourhood…

Let $K$ be a knot in an L-space. The knot $K$ is persistently foliar if, except for one meridional slope, every boundary slope of its complement is strongly realized by a co-orient…

Let $\mathcal F$ be a codimension one foliation on $\mathbb P^n$, with $n\ge 3$, and let $N^*_{\mathcal F}$ be its conormal sheaf. Let $\nabla_B$ be the Bott connection on…

Let $\mathcal F$ be a codimension one foliation on $\mathbb P^n$, with $n\ge 3$. Let $\Omega^1_{\mathcal F}$ denote its sheaf of foliated $1$-forms. A foliation is rational if it i…

Let $(X,\mathcal{F},t)$ be a projective klt adjoint foliated structure, and set … Assume that $K$ is pseudo-effective and $t<1$. Existence of good minimal models. The structure…

Trajectory-space invariance conjecture. The $v$-basic de Rham cohomology