5 problems

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…

Finiteness and uniqueness conjectures. The class $\alpha$ should be represented by at most finitely many connected embedded symplectic submanifolds; the stronger conjecture is that…

Let $n<9$ and consider the rational surface $\mathbb{CP}^2\# n\overline{\mathbb{CP}}{}^2$. A symplectic torus is an embedded symplectic surface of genus one, and two such tori are…

Symplectic surface representability conjecture. The class $A$ is represented by a connected $\delta0omega$-symplectic surface if and only if $[\delta0omega]\delta0cdot A>0$,…

Lê's conjecture. There exists a positive integer $N$, depending on $\alpha$, such that $N\alpha$ is represented by an embedded, not necessarily connected, symplectic surface.