14 problems

Let $X$ be a compact $4$-manifold, let $\Omega$ be a symplectic form on $X$, and let $\mathcal{P}$ be a constraint manifold defined by an almost-complex structure tamed by $\Omega$…

Let $S^6$ carry its standard almost complex structure $J$, and let $\operatorname{Aut}(TS^6)$ denote the automorphisms of its tangent bundle. For $\psi\in\operatorname{Aut}(TS^6)$,…

Yau's conjecture. For $n\geq 3$, any closed $2n$-manifold admitting an almost complex structure will also admit a complex structure.

Let $(X,J)$ be a closed almost complex $4$-manifold. Suppose there exists a closed $2$-form $\phi$ such that, for every $x\in X$, there is a closed connected pseudoholomorphic curv…

Let a geometric local curve element and an equivariant local curve element be local curve elements of the same topological type in the framework of the paper. Geometric–equivariant…

Complex–almost complex comparison conjecture. This map is an isomorphism. The conjecture is intended to permit almost complex methods to study invariants of complex threefolds, but…

Generic polygon image conjecture. The map $\alpha: \mathsf{TS}_{k}\rightarrow\mathsf{TP}_{k+6}$ has image in $\mathsf{TP}^{\mathsf{gen}}_{k+6}$ and is a homeomorphism onto its imag…

Let $M$ be a closed complex manifold of real dimension $2n$, and let the total Betti number of $M$ mean the sum of all its Betti numbers. Sullivan's conjecture. The minimal total B…

Let $M$ be a compact, simply-connected, nonspin four-dimensional almost complex manifold. An energy-minimizing biharmonic almost complex structure on $M$ determines a homotopy clas…

Let $(g,J)$ be a compatible almost Hermitian structure, and define … A biharmonic almost complex structure is a zero of $B$. Transversality conjecture. The map $B$ is transverse to…

Let $M$ be a compact 4-manifold, let $J$ be an almost complex structure on $M$, and let $H_J^-$ denote the $J$-anti-invariant part of the second cohomology, with…

Let $M$ be a compact 4-manifold and let $J$ be an almost complex structure on $M$. Denote by $H_J^-$ the $J$-anti-invariant part of the second cohomology and set…

Epsilon-regularity conjecture. There exist constants $\varepsilon,C,r_0>0$, depending only on $\Omega$, $J$, and $\sigma$, such that if

Let $(M,\Omega)$ be a compact symplectic four-manifold with $b^+(M)=1$, equipped with an almost complex structure $J$ tamed by $\Omega$. Let $g_{\Omega}$ be the metric determined b…