49 problems

Bryant's modified Bing–Borsuk conjecture. Every locally compact homogeneous ANR-space of dimension $n\geq 3$ is a generalized $n$-manifold. This modifies the Bing–Borsuk conjecture…

Let $\Gamma$ be a regular $(n+1)$-valent graph with a “good” $(\mathbb Z_2)^{n+1}$-coloring $\alpha$. A generalized $(n+1)$-skeletal expansion is an expansion denoted…

Let $(\Gamma,\alpha)$ be a finite connected regular $(n+1)$-valent graph whose edges are colored by $(\mathbb Z_2)^{n+1}$, with $\alpha$ a “good” coloring. A moment graph is the gr…

Let $M^n$ be a closed connected smooth $n$-manifold whose Stiefel–Whitney classes all vanish. A Riemannian metric is conformally flat when its Weyl tensor vanishes. Conformal-flat…

A closed symplectically aspherical manifold is a closed manifold whose symplectic form vanishes on every spherical homology class. The symplectically aspherical positive scalar cur…

Fix an error weight $t$, set $n=n_{\rm \min}(t)$, and consider the family of real minimal qubit permutation-invariant quantum error-correcting codes with parameters $(n,t)$. Real mi…

Let $\mathcal L\subset\mathbb{R}^n$ be an affine subspace satisfying Hypothesis D. Let $\mathcal M\subset\mathcal L$ be a smooth manifold which is non-degenerate when viewed as a m…

Codimension-two conjecture. For all sufficiently large even $n\in\mathbb{N}$,

Universal-cover homology conjecture. Then $M$ does not admit a metric with positive $m$-intermediate curvature. The conjecture is proposed as a common generalization of the aspheri…

Let $M$ and $N$ be closed orientable $n$-manifolds, and let $f:M\to N$ be a map of degree $\pm1$. Write $\operatorname{\mathsf{dcat}}$ for the distributional category of a manifold…

Let $\mathcal M$ be a non-degenerate submanifold of $\mathbb R^k$, and let $\psi:\mathbb N\to[0,1)$ be non-increasing. Let $W_k^\times(\psi)$ denote the set of points in $[0,1)^k$…

Let $\mathcal M$ be a bounded immersed submanifold of $\mathbb{R}^{M}$ with boundary, of dimension $n$ and codimension $R$. Suppose that $\mathcal M$ is $l$-nondegenerate for…

Let $\mathcal M$ be a bounded immersed submanifold of $\mathbb{R}^{M}$ with boundary, of dimension $n$ and codimension $R$. Suppose that $\mathcal M$ is $l$-nondegenerate everywhere…

Let ${\mathbb{Z}}_p$ denote the topological group of $p$-adic integers, and let $X$ be a connected $m$-manifold. Hilbert–Smith conjecture, $p$-adic formulation. The group…

Let $1\leq d<n$ be integers. Let $\widetilde{\mathcal M}_{n,d}$ be the class of submanifolds $\mathcal M\subset\mathbb R^n$ of dimension $d$ that are nondegenerate at every point,…

Let $X$ be a regular CW complex, and let $X^{(n)}$ denote its $n$-skeleton. Converse of the closed-manifold theorem. If $X=X^{(n)}$, then $X$ is a closed manifold. This would chara…

Let $n\geq2$ be an integer and let $M\subset\mathbb{R}^n$ be a manifold of finite type. Neighbourhood estimate conjecture. There exists a number $\sigma=\sigma(M)>0$ such that, for…

Let $n\geq 1$ be an integer, let $M\subset\mathbb{R}^n$ be a strongly non-degenerate analytic manifold, let $p>2$ be an integer, and let $D\subsetneq\{0,\dots,p-1\}^n$ be a choice…

Let $\mathscr{M}$ be an $m$-dimensional manifold in $\mathbb{R}^n$, with co-dimension $c=n-m$, and let $\mathrm{N}_{\mathscr{M}}(\delta,Q)$ be the counting function appearing in th…

Main Conjecture. The stated asymptotic holds under the proper curvature conditions above. This is presented as a basic open-ended conjecture motivated by the expected random distri…

Let $l\geq 2$ and $n\geq 2l$. A pair $(l,n)$ is called good when the conditions (I) and (II) defined earlier in the paper hold for the associated matrix construction. The induced m…

Let $\mathcal{M}$ be a $d$-dimensional submanifold of $\mathbb{R}^n$ in Monge form, with codimension $m=n-d$, and let $N_{\mathcal{M}}(Q,\delta)$ count rational points of denominat…

Spin torus-action conjecture. The manifold $M$ is homeomorphic to one of $S^3$, $S^4$, $S^5$, or

Let $\tau_{n,d}$ denote the threshold governing the dimension of the set of simultaneously Diophantine-approximable points on nondegenerate $d$-dimensional manifolds in…

Huang's conjecture. The asymptotic above should hold in the stated range of $\delta$.