9 problems

Sasakian–Einstein conjecture. Every such homotopy sphere admits a Sasakian–Einstein metric.

Unique-smooth-structure conjecture. These are the only values of $k$ for which $S^{k+1}$ has a unique smooth structure.

Boyer–Galicki–Kollár's conjecture. Every parallelizable exotic sphere admits a Sasakian structure.

Wang–Xu conjecture. The dimensions without exotic spheres are exactly $1,2,3$, possibly $4$, and $5,6,12,56,61$.

The method under discussion constructs exotic spheres with the stated symplectic-homological properties in dimensions $11$ and $15$. Conjecture. This method can be applied in any d…

Existence conjecture. Every odd-dimensional homotopy sphere which bounds a parallelizable manifold admits an Einstein metric with positive Einstein constant.

Let $M$ be a highly connected manifold admitting a positive scalar curvature metric and with vanishing Witten genus. A homotopy-sphere Ricci-positivity conjecture. There exists a h…

An exotic sphere is a smooth manifold homeomorphic but not diffeomorphic to a standard sphere. A Riemannian metric is strictly $1/4$-pinched in the global sense when its sectional…

Let $\Sigma^n$ be an exotic sphere, meaning a smooth manifold homeomorphic but not diffeomorphic to $\mathbb{S}^n$, and let $d$ be a positive integer. Mann's conjecture. If…