6 problems

Let $(X,\omega)$ be a symplectic $4$-manifold, and let $\Sigma\subset (X,\omega)$ be a smoothly embedded symplectic surface. Genus is measured among smoothly embedded surfaces repr…

Let $E(2)$ be the elliptic K3 surface, and let $X_{K_1}$ and $X_{K_2}$ denote the manifolds obtained from $E(2)$ by knot surgery using knots $K_1$ and $K_2$, respectively. Knot-sur…

Let $M$ be a closed smooth $4$-manifold. Uniqueness conjecture. $M$ admits Einstein metrics for at most one sign of the scalar curvature. The conjecture proposes that, for a fixed…

Universal-cover criterion. A smooth $s$-cobordism of elliptic $3$-manifolds is smoothly trivial if and only if its universal cover is smoothly trivial.

Let $Y$ be a homology $3$-sphere with $(Y)=0$, and let $\frak{ds}(Y)$ be finite. The bounding genus $|Y|$ is the invariant defined using homologically $1$-connected bounding $4$-ma…

Symplecticity conjecture. The manifolds $M_{10}$ and $M_{8}$ are symplectic and hence minimal.