13 problems

For each degree $d$, let $\mathbf{T}_{d}^{a}$ denote the count associated with the ellipsoidal superpotential for $\CP^2$ with parameter $a>1$. Monotonicity conjecture. The count…

Let $J$ be an almost complex structure on a manifold, and let $\alpha$ be a closed $J$-anti-invariant form, meaning that $J^*\alpha=-\alpha$. Draghici–Li–Zhang conjecture. The zero…

Symplectic invariance and Taubes correspondence conjecture. Once $f$ is of sufficiently high degree, $\mathcal{I}_{(X,f)}$ should be independent of $f$ and define a symplectic inva…

Local completeness conjecture. These canonical smoothings, including the curves obtained by smoothing arbitrary subsets of the nodes of $C_{\mathrm{nod}}$, comprise all possible sy…

Let $A_0$ and $A_1$ be real pseudoholomorphic curves in $\mathbb{CP}^2$, with each $A_k$ holomorphic for a tame complex-conjugation-invariant almost complex structure. Two such cur…

Generic almost-complex-structure conjecture. The same stratification can be achieved by a generic choice of almost complex structure $J$ on $X$.

Genericity conjecture. The required Whitney stratification and compatible system of tubular neighborhoods can be achieved by a generic choice of almost complex structure.

Let $d$ be an integer and let $p/q>1$ be a reduced fraction. The count $\mathbf{T}_{d}^{p/q}$ is the index-zero finite-energy holomorphic-plane count associated with degree $d$ cur…

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…

Eliashberg–Hofer conjecture. The differential satisfies $d^2=0$, and the homology $H_*(C_*,d)$ is independent of the contact form $\alpha$ for $\xi$ and of the complex structure…

Let $(M,c omega)$ be a symplectic manifold. Denote by $c \mathscr{J}(c omega)$ the space of almost complex structures on $M$ compatible with $c omega$, and let…

Let the assumptions of Lemma(c) hold, and consider the inequality in Remark. Equality conjecture. Under these assumptions, the inequality is an equality. This proposed equality wou…

Let $(M,\omega)$ be a closed symplectic manifold of real dimension at least four, and let ${\mathcal J}_{\operatorname{reg}}$ denote a subset of the space of smooth $\omega$-tame a…