40 problems

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…

Let $(X,g)$ be a globally hyperbolic spacetime, and let $\mathcal{N}$ be its space of null geodesics. For each point $p\in X$, write $\operatorname{sky}_p$ for its sky, viewed as a…

Let $L\subset(\mathbb{C}P^2,\omega_{\operatorname{FS}})$ be an embedded monotone Lagrangian two-torus, and let $\Lambda$ be the Legendrian lift of its canonical threefold Bohr–Somm…

Let $V$ be a symplectic vector space, let $X'\subset \P(V)$ be an irreducible Legendrian subvariety, let $g\in\mathfrak{sp}(V)$ be a nilpotent endomorphism, and let $m$ be an integ…

Let $X\subset \P^{2n-1}$ be an irreducible indecomposable Legendrian subvariety, and let $G<\P\operatorname{GL}_{2n}$ be a connected subgroup preserving $X$. Automorphism conjectur…

Let $\Gamma$ be a spatial graph, and let $S$ be the boundary of a tubular neighborhood of $\Gamma$. The spatial-graph DGA is the DGA constructed from a diagram of $\Gamma$ by the e…

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 $Y$ be the ambient space and let $\Lambda_n$ be the specified Legendrian stop. Write $\mathcal L^{\mathrm{emb}}_{\Lambda_n}(Y)$ for the moduli space of stops with embedded Lagr…

Let $\lambda$ be a Legendrian link, let $\varphi$ be a symmetry used to form the twist spun $\Sigma_\varphi(\lambda)$, and consider the folded cluster algebra associated with this…

Let $\lambda$ be a Legendrian link such that $\mathcal{M}_1(\lambda)$ has a cluster structure. Let $L_1$ and $L_2$ be exact Lagrangian fillings of $\lambda$, and let…

Let $L_1$ and $L_2$ be fillings, and let $d_\mu(L_1,L_2)$ be the smallest number of mutations relating them. Let $\Lambda(L_1,L_2)$ denote their Legendrian double. Mutation-distanc…

Let $\rho$ be the twisting monodromy, let $\rho^k$ denote its $k$-fold iterate, and let $\Sigma_{\rho^k}(\lambda(2,n))$ be the corresponding twist-spun Legendrian. Let $f(k)$ be th…

A Legendrian double is a Legendrian constructed from two fillings, and the mutation distance measures the smallest number of mutations relating the fillings. Mutation-distance conj…

A cubic planar Legendrian is a Legendrian surface in $J^1(S^2)$ arising from a planar trivalent graph; its dual graph has a chromatic polynomial. Treumann–Zaslow's conjecture. The…

Let $L$ be a linking disk at a Legendrian stop, and let the stop have Chekanov–Eliashberg differential graded algebra with coefficients in chains on its based loop space. Legendria…

O's conjecture. When $Q=\mathbb{R}^n$, the graded algebra $H^{\mathrm{string}}_*(\mathbb{R}^n,K)$ is isomorphic to the Legendrian contact homology of $\Lambda_K$ with coefficients…

Let $\Lambda(\mathbb{D}^2_{n,0,0,0})$ be the Legendrian link associated with a thrice-punctured disk with $n$ boundary marked points, with $n\geq4$, and consider the cluster pseudo…

Let $K$subset a contact manifold be a knot or link, let $\operatorname{Leg}(K)$ be the space of Legendrian embeddings of $K$, and let $\Lambda(\beta)$ be a maximal-Thurston--Benneq…

Let $\mathbb{S}^5$ be the unit sphere and let $\Sigma_H\subset\mathbb{S}^5$ be the hexagonal Legendrian minimal torus. For a surface $\Sigma\subset\mathbb{S}^5$, write…

Let $\mathbb{S}^5\subset\mathbb{C}^3$ be the unit sphere, and let $\Sigma_H\subset\mathbb{S}^5$ be the hexagonal Legendrian minimal torus. For a Legendrian surface…

Let $\Gamma\subset S^2$ be a cubic planar graph, let $L_\Gamma\subset J^1(S^2)$ be the associated Legendrian surface, and let…

Let $W$ be a stopped Weinstein manifold and let $\Lambda$ be a singular Legendrian submanifold in its boundary. Its Chekanov–Eliashberg dg-algebra is the dg-algebra associated to…

In a closed $5$-dimensional Sasakian manifold $N^5$, consider a solution of the weak Hamiltonian-minimal equation arising for a continuous conformal Legendrian map from a closed Ri…

Paracyclic lift conjecture. This simplicial $\infty$-category can be lifted to a paracyclic $\infty$-category.

Swappability conjecture. The Legendrian stops in Example $$ are swappable.