19 problems

Let $M$ be a closed positive scalar curvature $n$-manifold, and let $\widetilde M$ be its universal covering equipped with the metric lifted from $M$. The macroscopic dimension…

Let $M$ be a closed positive scalar curvature $n$-manifold with fundamental group $\Gamma$, and let $B\Gamma$ be a classifying space for its universal covering. The manifold $M$ is…

A Riemannian $4$-manifold is assumed to have uniformly positive scalar curvature. Gromov's macroscopic dimension conjecture. Its macroscopic dimension should be $2$. This conjectur…

Gromov's rational inessentiality conjecture. Then $M$ is not rationally essential.

Let $M$ be a closed positive scalar curvature $n$-manifold, and let $\widetilde M$ be its universal covering equipped with the metric lifted from $M$. The Gromov conjecture. Its ma…

Let $M$ be an open manifold of dimension $n$, where $n\ge 3$, equipped with a metric of uniformly positive scalar curvature. Its “dimension at large” is the macroscopic dimension,…

Refined macroscopic dimension conjecture. The inequality

Let $u$ be the solution of the parabolic Anderson equation with multiplicative noise, started from bounded initial data $u_0$. For $\beta$ as in the definition of the macroscopic H…

Let $M$ be a compact manifold of dimension $n$, let $\tilde{M}$ be its universal cover, and let $\dim_{mc}(\tilde{M})$ denote the macroscopic dimension of the universal cover. Let…

Let $X$ be a closed $n$-dimensional Riemannian manifold, and let $\operatorname{Lip}_1(X_{\neq 0}^{\to\circ})$ denote the metric space of distance-decreasing maps $X\to S^n$ of non…

Dranishnikov's conjecture. Every rationally inessential manifold $M^n$ is macroscopically small.

Gromov's conjecture. The inequality

Let $M$ be a closed oriented $n$-dimensional manifold, let $\pi=\pi_1(M)$, and let $B\pi$ be a classifying space. If $f\colon M\to B\pi$ classifies the universal bundle, call $M$ r…

Weak Gromov conjecture. The universal cover satisfies

Strong Gromov conjecture. The classifying map can be deformed into the $(n-2)$-skeleton $B\pi^{(n-2)}$.

Gromov's conjecture. The universal cover satisfies

Let $\pi$ be a group, let $B\pi$ be its classifying space, and let $H_n^{sm}(B\pi)$ denote the subgroup of homology classes represented by closed oriented $n$-manifolds whose unive…

Let $M$ be a closed $n$-manifold with a metric of positive scalar curvature, and let $\widetilde M$ be its universal cover. Weak Gromov conjecture. One has … This is a weaker form…

Gromov's macroscopic-dimension conjecture. Every closed Riemannian manifold with positive scalar curvature is md-small.