144 problems

Let $\Sigma_0$ and $\Sigma_1$ be homotopy spheres, and let $\Sigma_0\subset T^*\Sigma_1$ be a Lagrangian embedding. A bundle of rigid tubes over $\Sigma_1$ is a tube bundle of the…

Arnol'd's nearby Lagrangian conjecture. The Hamiltonian orbit of the zero section $L$ should be the set of all exact Lagrangians $L'$. This conjecture concerns the classification o…

$C^0$-flux conjecture. The Hamiltonian diffeomorphism group $\operatorname{Ham}(M,\omega)$ is $C^0$-closed in $\operatorname{Symp}_0(M,\omega)$.

Let $(V,\xi)$ be a closed contact manifold, where $\xi$ is a cooriented contact structure, and let $\operatorname{Cont}_0(V,\xi)$ denote the identity component of its contactomorph…

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…

Floer-theoretic identification conjecture. The Floer-theoretic coset is the same as the nearby Lagrangian torsion:

Let $f:X\to\mathbb{S}^2$ be a relatively minimal genus-$g$ Lefschetz fibration, where $g\geq 2$, and let its slope be … Here $K_f^2(X)=c_1^2(X)+8(g-1)$ and…

Let $(M,c9)$ be a closed symplectic manifold. Write $Hameo(M,c9)$ for the group of Hamiltonian homeomorphisms and $Sympeo_0(M,c9)$ for the identity component of the group of symple…

Joyce's conjecture. The holomorphic volume form $\Omega$ defines a stability condition in the Fukaya category of $X$.

Let $X$ be a minimal symplectic $4$-manifold, meaning that it contains no symplectically embedded exceptional sphere, and suppose that $X$ is not diffeomorphic to a ruled surface.…

Let $(M,\omega)$ be a closed symplectic manifold, let $L$ be a closed Lagrangian submanifold of $M$, and let $\varphi\in \operatorname{Ham}(M,\omega)$ be a Hamiltonian diffeomorphi…

Generation conjecture. The category $\mathcal{W}_\kappa(T^*Q)$ is split-generated by any $\kappa$-tuple of disjoint cotangent fibers.

Let $S_0,S_1\subset \mathbb{C}P^2$ be smoothly embedded symplectic surfaces representing the same homology class. Symplectic isotopy conjecture. The surfaces $S_0$ and $S_1$ should…

Let $f:X\to\mathbb{S}^2$ be a relatively minimal Lefschetz fibration, meaning that no fiber contains a $(-1)$-sphere. It is decomposable if it can be expressed as a fiber sum of tw…

Let $(M,\omega)$ be a closed symplectic manifold, let $k\geq 2$ be an integer, and equip the group of Hamiltonian diffeomorphisms with Hofer's metric. Hofer-distance conjecture. Th…

Let $N$ be a compact $C^infty$-manifold, let $\mathscr{X}=T^*N$, and let $I$ be an open interval containing $0$. For a Hamiltonian symplectic isotopy…

Let $L$ be a compact Lagrangian submanifold embedded in $cC^n$, and let its Maslov class be denoted by $\mu_L$. Maslov Class Conjecture. The Maslov class is nonzero: … This is pres…

Calabi extension conjecture. The Calabi homomorphism extends to a homomorphism

Let ${\mathcal P}^{ham}(Sympeo(S^2),id)$ denote the space of continuous Hamiltonian paths from the identity, let $\overline\mu^{path}$ be the extension of the homogeneous Calabi qu…

Let $Hameo(M,\omega)$ be the group of Hamiltonian homeomorphisms and let $Sympeo_0(M,\omega)$ be the identity component of the group of symplectic homeomorphisms. Proper subgroup c…

Let $(M,omega)$ be a closed symplectic manifold. The space $ham_\infty(M,\omega)$ consists of the continuous functions obtained as developing maps of continuous Hamiltonian paths,…

For compact Lagrangians $L,L'$ in a symplectic manifold $(M,\omega)$, let $B(L,L')$ be the supremum of the numbers $r a 0$ for which there exists a symplectic embedding … satisfyin…

Embedded contact homology invariance conjecture. One has $\partial^2=0$; $\widetilde{ECH}_*(Y,\lambda;\Gamma,G)$ is independent of $J$; and…

Let $(X,\omega)$ be a simply connected symplectic $4$-manifold. For the degree-$N_k$ symplectic branched covering associated with sufficiently large $k$, let $G_k^0$ be the relevan…

Let $(M,\omega)$ be a symplectic four-manifold equipped with toric fibrations, and let an almost toric fibration mean a fibration of $(M,\omega)$ obtained by allowing the nodal sin…