11 problems

Let $X$ be a space, let $NORM(X)$ be the site of normal coverings of $X$, and let $OPEN(X)$ be the standard site. Let $S$ be a set, write $S_{\#}$ for the associated constant coshe…

Let $E:\mathbf{BornCoarse}\to\mathbf{C}$ be a coarse homology theory, let $X$ be a uniform bornological coarse space, and let $i_\alpha:M_\alpha\to M$ be the canonical inclusions a…

Let $Y$ be a compact Hausdorff space and let $(Y_i)_{i\in\mathbb{N}}$ be a decreasing family of closed subspaces, with intersection given the subspace topology. Let…

Let $X$ and $Y$ be compact metric spaces with finite shape dimension, let $\mathfrak{a}:cX\to cY$ be a coarse map, and suppose that $M_X$ has $t\in\mathbb{N}$ ends. Choose…

Let $\mathbb{H}=\bigvee_{j\in\mathbb{N}}S^1_j$ be the Hawaiian earring with wedge point $x_0$, and let $c\mathbb{H}$ be its cone. Write $F_h=\pi_1(\bigvee_{j=1}^hS^1_j,x_0)$ for th…

Let $X$ be a compact metric space, and consider the finite approximations of $X$ constructed by the Main Construction, equipped with an hyperspace topology. Associate to the result…

Let $X$ be a locally compact separable metrizable space, and let $N_\alpha$ be the nerves of its open covers. Conjecture A. For each $n$, … This conjecture asserts the vanishing of…

Pro-homology conjecture. On the standard site $OPEN(X)$, if $X$ is Hausdorff paracompact, then

Let a flow have an exceptional minimal set, meaning a minimal set that is not a submanifold of the ambient manifold. The shape of a minimal set is related to the entropy of the flo…

Gupta–Mardia's conjecture. A new class of shape distributions should be constructible by this induction procedure, including a shape distribution of hypergeometric type…

Pro-fundamental-group Borel conjecture. If ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are equicontinuous, ${\mathcal A}_B$-like matchbox manifolds that have isomorphic…