30 problems

Let $f:\mathbb{C}\to\mathbb{P}^n(\mathbb{C})$ be an algebraically nondegenerate holomorphic curve, and let the hypersurfaces be in general position. Shiffman's conjecture. The tota…

Let $f:8?\mathbb{C}\to\mathbb{P}^n(\mathbb{C})$ be a holomorphic curve whose image spans a linear subspace of codimension $s$, and let $H_j$, $j=0,\ldots,q$, be hyperplanes as in C…

Wronskian generalization conjecture. In general, the Wronskian $W(f)$ should be replaced by a generalization of it, still of Wronskian type; if so, the expected truncation level is

A solenoid by holomorphic curves is a compact solenoid in the complex projective plane whose leaves are holomorphic curves. Holomorphic-solenoid conjecture. There exists a compact…

Green's conjecture. If $f$ omits $D$, then $f(\mathbf{C})$ is algebraically degenerate; equivalently, the conclusion of Green's theorem for finite-order holomorphic curves should h…

Let $X$ be a symplectic $4$-manifold with $b^+=1$ and $c_1^2>0$, and let $S\subset X$ be an embedded symplectic surface. Siebert–Tian conjecture. $S$ must be symplectically isotopi…

Let $B^{2n}(a)$ be the symplectic ball of capacity $a$, let $L$ be a Lagrangian submanifold in $B^{2n}(a)$, and let $J$ be an admissible almost-complex structure. Holomorphic-disk…

Let $\lambda$ range over partitions, let $c(\square)$ and $h(\square)$ denote the content and hook length of a box $\square\in\lambda$, and let ${\mathsf{W}}_{\lambda,\emptyset}$ b…

Let $(W,J)$ be the rational homology ball Stein filling of $L(9,2)$, and let $N_k\subset L(9,2)$ be the link shown in the paper's figure. The links $N_{3d}$ are considered for inte…

Let $m,n$ be positive integers, and let $G(m,m+n;\mathbb{C})$ denote the complex Grassmannian. A holomorphic 2-sphere is linearly full when it is not contained in a proper totally…

Let $m,n$ be positive integers, and let $G(m,m+n;\mathbb{C})$ denote the complex Grassmannian. A holomorphic 2-sphere is linearly full when it is not contained in a proper totally…

Let $m,n$ be positive integers, and let $G(m,m+n;\mathbb{C})$ denote the complex Grassmannian. A holomorphic 2-sphere in this Grassmannian has degree $\mathrm{d}$ and constant curv…

Let $m,n$ be positive integers, and let $G(m,m+n;\mathbb{C})$ denote the complex Grassmannian. A holomorphic 2-sphere in this Grassmannian has degree $\mathrm{d}$, defined as its a…

Let $\mathcal{M}(\alpha,\mu,\nu;\chi=\chi_0)$ denote the moduli space of punctured embedded curves with homology data $(\alpha,\mu,\nu)$ and Euler characteristic $\chi_0<1$. U(1)-s…

Generalized-quiver partition-function conjecture. After this substitution, the generalized-quiver partition function equals the generating function of Gromov-Witten invariants coun…

Let $K$ be a fibered knot, and write $F_K(x,q)$ as a series in $x$. Fibered-knot Laurent polynomial conjecture. For any fibered knot $K$, the coefficients of $F_K$ are Laurent poly…

Let $K:L^+\rightsquigarrow L^-$ be a Lagrangian cobordism, possibly immersed and non-monotone, and let ${CF^\bullet}(L^+,L')$ and ${CF^\bullet}(L^-,L')$ denote the relevant Floer c…

Let $M$ be a closed manifold of dimension at least $4$, and let $\omega=d^{\alpha}\lambda$ be an exact locally conformally symplectic form on $M$ whose Lee form $\alpha$ is rationa…

Let $(M,\omega)$ be a symplectic manifold, and let an almost complex structure $J$ on $M$ be super-rigid if every simple $J$-holomorphic map has non-negative index, every simple in…

Let $[i]$ be a closed geodesic whose corresponding SFT generator in $\mathbb{R}\times M$ is $q_i$. Consider a building configuration that may contribute to the differential of…

Let $[i]$ be a closed geodesic and let $q_i$ be the corresponding SFT generator in $\mathbb{R}\times M$. A building configuration is a holomorphic-curve building that may contribut…

Holomorphic Seifert–Weinstein conjecture. For any l.c.s. pair $(\omega,J)$ on $S^{2k+1}\times S^1$ or on $C\times S^1$ for $C$ a threefold, there is an elliptic, non-constant $J$-h…

Let $(M,\omega)$ be an integral symplectic manifold, meaning that its symplectic form represents an integral cohomology class. The Biran conjecture. All obstructions to symplectic…

Let $(M,\omega)$ be a symplectic manifold and consider symplectic embeddings of standard balls into $M$. An obstruction to such an embedding is a condition preventing the embedding…

The all-values existence conjecture. For fixed $m$, holomorphic solutions with constant curvature in $G(m,n)$ can be constructed for every integer $r$ satisfying