286 problems

Let $Y$ be a closed manifold and let $\lambda$ be a contact form on $Y$. Its Reeb vector field is the vector field $R$ determined by $\lambda(R)=1$ and $d\lambda(R,\cdot)=0$; a Ree…

Let $V$ be a normal affine variety of complex dimension $n$ with an isolated singularity at $0$ that is numerically $\mathbb{Q}$-Gorenstein. Write $\operatorname{min.disc}(V,0)$ fo…

Let $\alpha$ be a contact form inducing the standard contact structure $(S^3,\xi_{st})$, and let $\Lambda\subset S^3$ be a Legendrian submanifold. Arnold's chord conjecture. There…

Low's conjecture. Events $x,y\in X$ are causally related if and only if their skies $S_x,S_y$ are linked in $\mathcal N_X$.

Let $(M,Z,\alpha)$ be a compact $b^m$-contact manifold, with critical set $Z$. A singular periodic orbit is an integral curve of the Reeb $b$-vector field in $M\setminus Z$ whose l…

Let $(Y,\xi)$ be a closed contact manifold, let $\alpha$ be any contact form on $Y$, and let $\Lambda\subset Y$ be a closed Legendrian submanifold. Arnol'd's chord conjecture. The…

Let $Y$ be a star-shaped hypersurface in $\mathbb{R}^4$, and call a periodic orbit simple if it is not a multiple cover. Hofer-Wysocki-Zehnder's two-or-infinity conjecture. A Hamil…

Let $Z$ be a complex projective manifold equipped with a contact structure, namely a holomorphic hyperplane distribution $D\subset TZ$ such that the induced bundle morphism … is ev…

Let $(Y,\xi)$ be a contact $3$-manifold. A Stein filling of $(Y,\xi)$ is a Stein domain $(N,\omega)$ with boundary contact manifold $(Y,\xi)$, and write $b_1(N)$, $\chi(N)$, and…

Finiteness conjecture. Every planar contact $3$--manifold should admit only finitely many deformation classes of minimal symplectic fillings.

Let $M$ be a globally hyperbolic spacetime, let $\mathcal{N}$ be its manifold of cone geodesics, and for each event $x\in M$ let … be its sky. Low's sky-linking conjecture. Causal…

Loose Legendrian rigidity conjecture. The Legendrian $\varphi_\infty(\Lambda)$ is still loose.

Let the spacetime be $(3+1)$-dimensional and globally hyperbolic, with a Cauchy surface homeomorphic to $\mathbb{R}^3$. The skies of events are viewed as Legendrian links in the re…

Let $K$ be a knot or link type, let $\chi(K)$ be its maximal Euler characteristic, and let $\operatorname{SL}(K)$ be the maximum self-linking number of its braid representatives. D…

Let $(M_k,\ker\alpha_k)$ be an orderable contact manifold, and let $c_+^{\alpha_k}$ be the associated contact spectral invariant. A map is $\alpha_k$-spectral when it satisfies the…

Let $(M,\zeta)$ be a closed contact $3$-manifold, where $\zeta$ is a contact distribution, and consider the Chern-Hamilton energy functional on the space of compatible metrics for…

Let $R$ be a Reeb vector field on a closed $3$-manifold. Two-or-infinitely-many periodic-orbits conjecture. The vector field $R$ has either two or infinitely many periodic orbits.…

Let $D$ be a circular spherical divisor, meaning a topological divisor consisting of a cycle of spheres, and let $b^+(D)$ denote the positive index of its intersection form. The as…

The affine uniruledness conjecture.

Let $(M,\xi)$ be a contact 3-manifold equipped with a compatible metric $g$. Write $Ricci(g)>0$ for positivity of its Ricci curvature, and let $(\mathbb{S}^3,\xi_{std})$ denote the…

Stipsicz's conjecture. Every closed contact $3$-manifold admits a universal bound on the signatures and Euler characteristics of its possible Stein fillings.

Let $(M,g)$ be a causally simple spacetime, and let $\mathcal{N}_g$ denote its space of null geodesics. A conformal embedding is an embedding of $(M,g)$ into another spacetime that…

Large minimal Chern number conjecture. If $c_B$ is sufficiently large, for example in the range suggested by the source, then a dynamically convex contact form on $M$ carries infin…

Planar torsion vanishing conjecture. If $(M,\xi)$ has planar torsion of any order, then its untwisted Ozsváth–Szabó contact invariant vanishes; moreover, the twisted invariant vani…

Self-linking classification conjecture. Negative torus knots are determined by their self-linking number; equivalently, two transversely represented negative torus knots of the sam…