31 problems

Asymptotic preservation conjecture. For general hyperkähler manifolds, generic asymptotically conical or asymptotically locally conical behaviour will be preserved under modificati…

Let $(M,g)$ be a gravitational instanton. Finite-topological-type conjecture. The manifold $M$ has finite topological type, meaning that it can be compactified into a smooth manifo…

Let $(M,g)$ be a gravitational instanton. Asymptotic-cone uniqueness conjecture. The gravitational instanton $(M,g)$ has a unique asymptotic cone. Uniqueness is known for complete…

A triple $(a,b,c)$ extends when it occurs as the first three row lengths of a P-position $(a,b,c,d)$ of $4\times n$ Chomp. Let $\mathrm{width}(c)$ denote the width of the band of e…

Let $\mathcal{X}$ be an unbounded metric space. Call $\mathcal{X}$ large if its asymptotic space in some (or every) nonstandard model $M$ contains a subspace isometric to…

The STCMC-foliation and the CMC-foliation are foliations used to define centers of mass in general relativity, with STCMC denoting spacetime constant-mean-curvature foliations and…

Asymptotic conicity conjecture. A Kähler-Ricci shrinker is weakly asymptotically conical in the sense that it has a unique asymptotic cone at infinity, which is a Kähler cone metri…

Compactification conjecture. For any $\kappa_1>\lambda$ and $\kappa_2>\mu$, $(X,I,\omega,\Omega)$ can be compactified complex analytically to a weak Fano manifold. Furthermore, the…

Let a symmetric photon surface in an asymptotically flat spacetime have a part tending towards spatial infinity, $\rho\to\infty$, where $h\to 1$ as $\rho\to\infty$. Cederbaum–Jahns…

For each integer $n\geq 2$, let $\mathcal{C}_n$ be the $n$-dimensional cube, let $\mathcal{C}_n^*$ be the $n$-dimensional cross-polytope, and let $d_{BM}$ denote the Banach–Mazur d…

Quantified asymptotic Euclideanity conjecture. There exists $\delta(N,\epsilon)>0$ such that, for every $\delta\in(0,\delta(N,\epsilon)]$, there exists $R_0\geq 1$ for which…

Let $g_{L^2}$ be the natural complete hyperkähler metric on the moduli space of Higgs bundles, and let $g_{\mathrm{sf}}$ be the semiflat metric arising from the algebraic integrabi…

Let $X_5(\theta)$ be the hyperpolygon space associated with a generic parameter $\theta$. A smooth noncompact hyperkähler manifold is quasi-asymptotically locally Euclidean when it…

Let $\operatorname{Bun}_n$ be the moduli stack of bundles over the $n$th infinitesimal reduction of the curve, let $\mathcal S_{cusp}(\operatorname{Bun}_n)_\chi$ be the cuspidal su…

Zong's asymptotic conjecture. The sequence of covering-function values satisfies

Let $Y$ and $Z$ be unbounded metric spaces, and let $\tilde{r}=(r_n)_{n\in\mathbb{N}}$ be a scaling sequence. For a metric space $W$, write $Seq(W,\tilde{r})$ for its sequence spac…

Let $(X,d)$ be an unbounded metric space and let $\tilde{r}=(r_n)_{n\in\mathbb{N}}$ be a scaling sequence. Define … for $\tilde{x},\tilde{y}\in Seq(X,\tilde{r})$; this is a pseudom…

Yang–Zagier–Zhang conjecture. The error term in this asymptotic admits an expansion in $g^{-1}$ whose leading term is $-\frac{\pi^2}{144}\frac{1}{g}$, with explicit coefficients fo…

Let $g,n$ be nonnegative integers satisfying the assumptions in the conjecture, and let $d_1,\dots,d_n$ be nonnegative integers with … Write…

Linear-bound conjecture. The compactness constants satisfy

Gaussian Betti-number conjecture. Fix $q\in[1,n]$. There is a normalizing function $F_q(d)$, depending on $X$ and geometric data, such that for every sequence of integers $p_d$ sat…

Asymptotic vanishing conjecture. Fix $q\in[2,n]$. In the situation of the nonvanishing theorem referenced in the source, there is a constant $C_3$, depending on $X$, $A$, $B$ and…

Asymptotic one-sided kissing-number conjecture.

Let $\mathbf{K}$ be an $\mathbf{o}$-symmetric $d$-dimensional convex body, and let $\operatorname{ill}(\mathbf{K})$ be its illumination parameter. Quantitative illumination conject…

Let $\mu$ be a measure under consideration, and let $\mathcal{S}_{nt}^{sing}(\mu)$ denote its non-tangential singular set. Meagreness conjecture. The set…