263 problems

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…

A flow is codimension one when either its weak-stable or weak-unstable foliation has codimension one. Verjovsky's conjecture. Every smooth codimension-one Anosov flow on a manifold…

Let a foliation on a variety over a finitely generated base be reduced modulo almost all primes, and suppose it is closed under $p$-th powers modulo $p$. Ekedahl–Shepherd-Barron–Ta…

Let ${\mathcal W}$ be a non-parallelizable $3$-web on a real or complex $2$-dimensional manifold. A $3$-web is linearizable if it is equivalent, up to a local diffeomorphism, to a…

Let $X$ be a rationally connected manifold, and let $\mathcal{F}\subset T_X$ be a regular foliation. Touzet's conjecture. The leaves of $\mathcal{F}$ are algebraic. This conjecture…

A diffeomorphism $f:M\to M$ is dynamically coherent partially hyperbolic when its partially hyperbolic structure admits invariant center foliations. It is plaque expansive if there…

Let $(G,g)$ be a semi-Riemannian Lie group with a subgroup $K$ generating a left-invariant conformal foliation $\mathcal F$ on $G$ of codimension two. Semisimple subgroup minimalit…

Let $M_g^2$ be a surface in ${\mathbb T}^3$, let $B$ be a direction, and let ${\cal F}_B(M_g^2)$ denote the corresponding foliation or section system. Call ${\cal F}_B(M_g^2)$ chao…

Let $(M,\gamma)$ be a taut sutured manifold, meaning a sutured manifold satisfying the tautness condition, and suppose that $\chi(M)<0$. A finite-sheeted cover is a covering … with…

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

Let $M$ be a hyperbolic $3$-manifold, let $\mathscr F$ be a taut foliation of $M$, and let $\widetilde{\mathscr F}$ be its lift to the universal cover. For a leaf $\lambda$ of…

Let $X$ be a rationally connected projective variety over an algebraically closed field of characteristic zero, and let $\mathcal F$ be a regular foliation on $X$. Algebraic integr…

Let $\alpha\in H_{n-1}(\mathbb{T}^n,\mathbb{R})$ be a totally irrational class. For rational classes $r_k$ converging to $\alpha$, let $\Delta W_{r_k}=\omega(r_k)-S(r_k)$. Continui…

Recall that $\overline{\mathrm{B}\Gamma}^r_k$ classifies codimension $k$ $C^r$-Haefliger structures with trivial normal bundle. Haefliger–Thurston's conjecture. The space…

Let $(M,\mathcal{F})$ be a foliation on a compact manifold, let $BG$ be the classifying space of its holonomy groupoid, and let $\mathcal{L}^{lf}$ be the cosheaf assigning to each…

Let $M$ be a closed simply connected manifold admitting a Riemannian metric of nonnegative sectional curvature. Grove's conjecture. Then $M$ has a DDBD structure, that is, an isopa…

Classical L-space knot conjecture. The knot $kappa$ is persistently foliar if and only if it has no non-trivial L-space or reducible surgeries.

Let $M$ be a compact manifold and let several closed $1$-forms on $M$ be given. Their common level is the intersection of the corresponding level sets at a generic choice of levels…

Let $\frac{r_*}{s_*}:=(e_0=\frac{r_0}{s_0},\frac{r_1}{s_1},\ldots,\frac{r_n}{s_n})$ be an $n+1$-tuple, let $J\subset\{1,\ldots,n\}$, and let $J$-realizability mean the existence of…

Conjecture 6.1. If $\frac{1}{2}p_1(F)=0$ and the Ricci curvature of $g^F$ along each leaf is positive, then the Witten genus

Thurston's uniformity conjecture. Every $\mathbb{R}$-covered foliation of $M$ should be uniform.

Let $(M,\gamma)$ be a sutured 3-manifold admitting a taut, transversely oriented depth one foliation $\mathcal{F}$. Write $[\mathcal{F}]\subset H^{1}(M;\mathbb{R})$ for the foliate…

Questions on local orientations and foliations. The paper asks: (i) to what extent a local orientation in which every oriented loop is homotopically essential can be extended over…

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…