4 problems

Singular nilpotent fibration conjecture. A singular nilpotent fibration exists on a collapsed manifold of curvature bounded below and with local bounded covering geometry.

Aspherical limit conjecture. Then $X$ is a differentiable manifold in the Ricci-curvature setting, or a topological manifold in the sectional-curvature setting, and is aspherical.

Let $\Gamma_{\varepsilon}(x)$ be the fibered fundamental group, namely the image of $\pi_1(B_{\varepsilon}(x))\to\pi_1(B_1(x))$ induced by inclusion. RCD fibration conjecture. Ther…

Let $M^n$ be a closed smooth $n$-manifold. Write $\operatorname{MinVol}(M)$ for the infimum of $\operatorname{vol}_g(M)$ over all smooth complete Riemannian metrics $g$ on $M$ whos…