10 problems

SYZ singularity conjecture. The limiting manifold should have singularities in codimension two, with monodromy transformations defined for each loop about the singular set.

Let $N$ be a compact and complete affine manifold, and let $\widetilde{N}\to N$ be a finite Galois cover. Affine-map conjecture. The cover $\widetilde{N}$ is the domain of a nontri…

Franks's conjecture. The diffeomorphism $f$ is $C^0$-conjugate to a hyperbolic infranilautomorphism.

Let $M$ be a closed complete affine manifold, and let $\pi_1(M)$ denote its fundamental group. A group is word hyperbolic if it is hyperbolic in the sense of Gromov. Affine-manifol…

Let $M$ be a closed affine manifold, and let $\|M\|$ denote its simplicial volume. Simplicial-volume conjecture. The simplicial volume of $M$ vanishes: … This is a stronger form of…

Let a family of Calabi–Yau manifolds converge to a large complex structure limit. Kontsevich–Soibelman modified Strominger–Yau–Zaslow conjecture. The manifolds collapse to affine m…

Let $M$ be a closed manifold supporting an affine structure, meaning a closed manifold admitting an atlas whose transition maps are restrictions of affine isomorphisms. Let $\|M\|$…

Let $M$ be a closed aspherical tangentially flat manifold, meaning a closed aspherical manifold whose tangent bundle is flat. Let $\chi(M)$ denote its Euler characteristic. The asp…

Vanishing simplicial volume conjecture. If $M$ is a closed manifold supporting an affine structure, then

Let $B$ be an affine manifold serving as the base of a Lagrangian torus fibration, with the associated torus fibration locally modelled on affine space. A Legendre dual constructio…