24 problems

Bressan's compactness conjecture. The sequence $(X_n)$ is strongly precompact in $L^1_{\mathrm{loc}}(I\times\mathbb R^d;\mathbb R^d)$. This conjecture concerns compactness of flows…

Let $M$ be complete, let $O$ be a fixed origin, and suppose that for some $r_0>0$ the Ricci curvature is bounded below by $(n-1)/(4r^2)$ whenever $r\geq r_0$. Higher-dimensional Ri…

Compactness–sequential compactness conjecture. For first countable perfectly normal spaces, compactness and sequential compactness should coincide.

Let a sequence of sets in $\mathbb{R}^{n}$ consist of smoothly embedded connected submanifolds without boundary and have uniformly bounded curvature everywhere. Consider their lift…

Let $\Omega\subset\mathbb{R}^N$ be the domain under consideration, let $p>1$, and let $\{u_n\}\subset C^1(\overline{\Omega})$ satisfy, in the weak sense, for some positive constant…

Compactness conjecture. Is $Min(Q)$ compact?

Let $(X,d)$ be an infinite ${\bf US}$-space, meaning an ultrametric space generated by a labeled star graph with a center $c$. Put … and let $d_0$ be the restriction of $d$ to…

Compactness and blow-up conjecture. If $1+2\mathsf{k} \leq n \leq 20+2\mathsf{k}$, then the full solution set of the constant $Q^{(2\mathsf{k})}$-curvature problem is…

Compactness characterization conjecture. The following statements are equivalent: $(X,d)$ is compact; and there is a labeled ray $R(l)$ generating $(X_0,d_0)$ such that

Let $\{a,b\}$ be the discrete topological space with two points, and let $\mathbb R$ be the real line with its usual metric. For a class $P$ of morphisms, write $P^{lr}$ for its do…

Let $(M_i,g_i)$ be a sequence of Riemannian manifolds with non-negative scalar curvature. Assume that their volumes and diameters satisfy uniform upper bounds. Gromov's compactness…

Let $\{u_\delta\}_{\delta\in(0,1)}$ be a sequence with $u_\delta\in \mathcal{S}^{\bm\nu}_{w_\delta}(\mathbb{R}^d)$, where the kernels $w_\delta$ satisfy the kernel assumptions in t…

Let $T$ be a rayless countable tree, meaning that $T$ contains no ray. For a vertex labeling $l\colon V(T)\to\mathbb{R}^{+}$, let $d_l$ be the generated ultrametric. Compact-case a…

Hayut's conjecture. $\kappa^+$-$\Pi^1_1$-subcompactness and $\kappa$-compactness are equiconsistent.

The strong compactness result concerns a family of coefficients with a time regularity condition expressed through a uniform bound in $W^{s,1}_t W^{-N,p}_x$. Time-regularity conjec…

Let $\textup{\textsf{HBU}}$ denote the Heine–Borel theorem and let the fragment of $\textup{\textsf{NFP}}$ be the one described in Theorem cited in the source. A strong $\mu N$-red…

Let $(\Gamma, {\preccurlyeq})$ be a totally ordered set and let $(X, \tau)$ be a $\Gamma$-ultrametrizable topological space, meaning that $\tau=\tau_{d_{\Gamma}}$ for some ultramet…

Compactness conjecture for minimizing sequences. There is a sequence of Möbius transformations $G_j$ of $\mathbb{R}^3$ such that the surfaces $G_j\circ\mathbf{x}_j(K)$ can be parti…

Equiconsistency conjecture. The existence of $\textbackslash kappa$ which is $\textbackslash kappa$-compact is equiconsistent with the existence of a cardinal…

Let $M$ be the underlying manifold, and let $(\varepsilon_i,\Psi_i,{\bm\xi}_i,A_i)$ be a sequence of solutions of the blown-up ADHM$_{1,k}$ Seiberg--Witten equation satisfying … wi…

Let $G$ be a compact Lie group and let $B^m$ denote the $m$-dimensional ball. For $m\geq 6$, consider the space ${\mathcal A}_G(B^m)$ of weak connections. Weak sequential closednes…

Let $\Omega$ be the domain, let $p$ be the exponent, let $m\geq 1$, and let ${\bf v}(x,t)$ be the solution under consideration. Let $\Upsilon_p$ denote the exponential rescaling ra…

A convex metrizable set is a convex set equipped with a metrizable topology, and a convex set is $\mu$-compact when it satisfies the $\mu$-compactness condition used in the Vesters…

Let $\Lambda\in\mathcal M_f([0,1])$, and let $\mathcal V^\Lambda_t$ be the tree-valued $\Lambda$-resampling dynamics at time $t$. Write $\mathbb U_c$ for the space of compact metri…