138 problems

LeBrun–Salamon conjecture. The manifold $Z$ must be homogeneous.

Let $(W^4,\vlambda,\vphi)$ be a $4$-dimensional Weinstein domain, and let $L\subset W$ be a properly embedded Lagrangian disk with Legendrian boundary…

Let $(M,\omega)$ be a symplectic manifold and let $H:M\to\mathbb{R}$ be a Hamiltonian. An energy level is a level set $H^{-1}(c)$, and a hypersurface $Y\subset M$ is of contact type…

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 $(M,Z,\alpha)$ 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…

Gompf's conjecture. No nontrivial Brieskorn sphere, with either orientation, arises as the boundary of a Stein domain in $\mathbb{C}^2$.

Let $k \ge 1$ and $n \ge 2$ be integers, and consider closed $(2n+1)$-dimensional contact manifolds, as well as algebraic torsion and untwisted algebraic torsion in that dimension.…

Let $\Sigma$ be a smooth manifold, let $ST^{\ast}\Sigma$ denote its unit cosphere bundle, and let $(f_t)_{t\in\mathbb R}$ be a positive path of contactomorphisms of…

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

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 $D$ 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…

Nonexistence conjecture. There are no smooth absolute minima of $E$ of degree greater than $1$, irrespective of the coupling constant.

Let $v$ be a non-vanishing (traversing ?) vector field on a compact $(2n+1)$-dimensional manifold $X$, and let $\xi$ be a $(2n)$-dimensional distribution normal to $v$ that admits…

Virtual Legendrian isotopy conjecture. Two Legendrian links in $ST^*S^m$ or $ST^*\mathbb C P^n$ are Legendrian isotopic if and only if they are virtually Legendrian isotopic. The s…

Strong virtual Legendrian unlinking conjecture. There is no non-negative strong allowable virtual Legendrian isotopy of one component of $L$ to the other.

Strong virtual Legendrian isotopy conjecture. There exists no strong allowable positive, or non-constant non-negative, virtual Legendrian isotopy of the sphere fiber of $ST^*M$ to…

Let $S^{2n+1}$ denote the $(2n+1)$-dimensional sphere, and let a contact manifold mean a manifold equipped with a contact structure. Sphere overtwistedness conjecture. The sphere…

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

Finiteness conjecture. The set $\mathcal{C}_{(M,\xi)}$ is finite.

Let $\gamma_t$, $t\in[0,1]$, be an eversion of a smooth circular wave front with no dangerous self tangencies: a family of generic oriented wave fronts in $\mathbb{R}^2$ whose endp…

Let $(M,g)$ be a globally hyperbolic $(3+1)$-dimensional spacetime with a Cauchy surface diffeomorphic to a subset of $\mathbb{R}^3$, and let $N$ be its manifold of light rays. For…

Let $(M,g)$ be an orientable, time-oriented, globally hyperbolic $(2+1)$-dimensional spacetime with a Cauchy surface $\Sigma\subset M$ diffeomorphic to a subset of $\mathbb{R}^2$.…

Palis-type conjecture. The space of closed oriented hypersurfaces in a contact manifold $(Y,\xi)$ decomposes into two open and $C^2$-dense sets: the set of convex hypersurfaces and…