137 problems

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…

Let $(X,\mathcal{L},\alpha)$ be an oriented $-1$-shifted contact derived stack over $\mathbb{K}$, and let $\varphi\colon L\to X$ be a proper oriented Legendrian. Write … for the re…

Let $\Lambda \subset (\Sigma \times \mathbb{R}, \xi_\Gamma)$ be a Legendrian knot. Let $Y(\Sigma, \Gamma) \subset \Sigma \times \mathbb{R}$ be the convex sutured contact manifold a…

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…

Let $Y\subset \mathbb{R}^4$ be a hypersurface with first Betti number $b_1(Y)=0$, and let $W$ be the domain bounded by $Y$. The domain $W$ is rationally convex if it satisfies the…

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 $(Y,\xi)$ be a contact 3-manifold, and consider the half-symplectization $Y\times(-\infty,0]$ with symplectic form $d(e^t\alpha)$, where $\xi=\ker\alpha$. A compact properly em…

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 $(M,\omega)$ 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…

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

Legendrian Gauss-map rigidity conjecture. The diagram relating the Lagrangian Gauss maps of $\Lambda$ and $\varphi_\infty(\Lambda)$ commutes:

Let $k,n,\ell$ and $d$ be as in the construction of the twist-spun torus link, and let $\Sigma_{\psi^\ell}(\Lambda(k,n-k))$ denote that twist-spun link. An orientably exact Lagrang…

Let $M$ be the configuration manifold, let $L$ be the corresponding magnetic Lagrangian, and let $\Sigma_{\kappa}$ be the energy hypersurface at energy $\kappa$. The lowest Mañé cr…