67 problems

Let $X$ be a closed $2n$-manifold and let $\omega$ be an integral symplectic form on $X$, meaning that its cohomology class lies in the image of…

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…

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…

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…

Legendre-transform equivalence conjecture. The natural map induced by the Legendre transform,

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…

Let $N$ be a $3$-manifold and let $S^{1} \times N$ admit a symplectic structure $\omega$. Write the Künneth component of $[\omega]$ in $H^{1}(N,\mathbb R)$ as the corresponding cla…

Let $N$ be a $3$-manifold, and let $S^{1} \times N$ admit a symplectic structure $\omega$. Taubes's conjecture. Then $N$ admits a fibration over $S^{1}$. This conjecture relates sy…

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

Nullhomotopy-dependence conjecture. The classes $\tau_{2k}(L,\gamma_t^i)$, for $i=1,2$, differ by $p_k(E)$.

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 $K$ be an orientable polyhedral Lagrangian in $(\mathbb{R}^4,\omega)$, let $v$ be a valence-$4$ vertex, and let $\Gamma_{v,\delta}\subset (S^3,\xi_{\mathrm{std}})$ be the Legen…

Let $L\subset M$ be a Maslov $0$ Lagrangian torus, and let $C^*(L,M)$ be its Floer complex with differential satisfying … Here $\tau(L)$ denotes the relevant boundary-depth invaria…

Veronese projection conjecture. The Lefschetz bipencil constructed in this section is isomorphic to the composition

Let $W$ be the completion of a Liouville domain, and let $\overline{\operatorname{Cont}}_{0,c}(W\times S^1)$ be the $C^0$-completion of the identity component of the group of compa…

Let $L$ be a displaceable Lagrangian in a symplectic manifold $M$, and let $d_{C^0}$ be the $C^0$ distance on Hamiltonian diffeomorphisms. Local Hamiltonian nondisplacement conject…

Formality conjecture. The Floer--Fukaya $A_\infty$ algebra

Let $M$ be a compact or Liouville symplectic manifold, and let $L\subseteq M$ be a compact, relatively exact Lagrangian. Let $\tilde\phi:L\to L$ be a diffeomorphism extending to a…

Let $N$ be a manifold, let $T^*N$ carry the cotangent-bundle structure, and let … DrT^N={(q,p)in T^N:|p|leq r}. … is bounded on the space of exact Lagrangians contained in…

Let $L$ be a closed smooth manifold and let $T^*L$ denote its cotangent bundle. For every compactly supported Hamiltonian diffeomorphism $\phi$ of $T^*L$, consider the intersection…

Let $W$ be a convex-at-infinity symplectic manifold. For each contact isotopy $\varphi_t$, let $\mathrm{HF}(\varphi_t)$ denote its Floer cohomology, and let $\mathrm{SH}^{e}(W)$ be…

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