49 problems

Let (KE) denote the energy-comparability property, and suppose a metric measure space admits a heat kernel satisfying the heat kernel estimate (hk). Heat-kernel implication questio…

Cheeger's chart-image conjecture. If $(U,\phi)$ is an $n$-dimensional chart, then

For the Sierpiński carpet, let $\alpha_1$ denote its critical Besov exponent, let $d_h$ be its Hausdorff dimension, and let $d_{th}$ be its topological Hausdorff dimension. Sierpiń…

Let $(X_n,d_n,\mu_n)$ for $n\in\mathbb{N}$ be a sequence of metric measure spaces that converges to $(X,d,\mu)$ in the Gromov–Wasserstein distance. The associated multidimensional…

Let $(M^{n},g,f)$ be a complete smooth metric measure space, and let $lambda_i$ be the $i$-th eigenvalue, for $i=1,2,\ldots,k$, of the drifting-Laplacian eigenvalue problem conside…

Let $(M_i,p_i)$ be $n$-dimensional complete pointed Riemannian manifolds with a uniform lower Ricci-curvature bound, and suppose that a subsequence converges in the pointed measure…

Let $(X,d,\mu)$ be a measured Gromov--Hausdorff limit of $n$-dimensional compact Riemannian manifolds $M_i$ with $\operatorname{Ric}_{M_i}\ge (n-1)k$, equipped with normalized Riem…

Let $(X,r,\mu)$ be a Polish metric measure space, meaning that $(X,r)$ is Polish and $\mu$ is a probability measure on its Borel sets. Gromov's $\underline{\square}_1$-metric is ob…

Metric-measure and Heisenberg-group conjecture. Problem $$ should yield the same result when considered over a metric measure space or a Heisenberg group.

Kitagawa–Letrouit's quantitative stability conjecture. The quantitative stability results for optimal transport are also true in more general metric measure spaces with synthetic c…

Let $\left(X,d_t,\mathfrak{m}_t\right)_{t\in I}$ be a family of metric measure spaces that is a non-collapsed weak/rough Ricci flow, meaning that it is both a weak/rough sub-Ricci…

Grigor’yan–Hu–Lau resistance conjecture. The form $(\mathcal{E},\mathcal{F})$ satisfies $\textup{HKE}(\beta)$ if and only if it satisfies both $\textup{PI}(\beta)$ and…

Let $(X,d,m)$ be a metric measure space satisfying volume doubling, denoted by. Let denote the relevant Poincaré inequality, the uniform capacity condition, and the two-sided heat-…

Let $(\mathfrak{X},d,\mu)$ be a metric Polish probability space. Equivalence conjecture. The space $(\mathfrak{X},d,\mu)$ is asymptotically average-case approximable if and only if…

Let a metric measure space carry a $p$-energy with $p$-walk dimension $\beta_p$, and let its associated $p$-energy measure be defined relative to the underlying measure. Singularit…

Metric-measure-space extension conjecture. The results should also hold in more general metric measure spaces with synthetic curvature bounds.

Let $m$-rectifiable measures $\mu$ be measures on $\mathbb{R}^d$ that are $m$-rectifiable, and let $e_{N,p}(\mu)$ denote the $N$-point quantization error for exponent $p$. Graf–Lus…

Let $X$ be an $n$-dimensional projective variety with log terminal singularities, and let $\theta_X$ be a smooth Kähler metric on $X$. Let $(X,\omega)\in\mathcal{RK}(n)$, so that…

Let $X$ be an essentially non-branching metric-measure space, and let $C(X)$ denote its Euclidean $N$-cone. Euclidean cone curvature-dimension conjecture. The Euclidean $N$-cone…

Let $B$ be a metric-measure base of dimension $d$, let $F$ be an essentially non-branching metric-measure space, and let $f$ be a Lipschitz nonnegative warping function. Let…

Let $(\mathsf{M},l,\mathfrak{m})$ be a metric measure spacetime. It is infinitesimally Minkowskian if, for all $l$-causal functions $u$ and $v$, its maximal weak subslope satisfies…

A globally hyperbolic finite-dimensional TCBB space $(\mathsf{M},l)$ is a metric measure spacetime with synthetic timelike sectional curvature bounded from below. Let $I^+(o)$ deno…

Let $(X,d,cmu)$ be a metric-measure space, and let the sample size increase for the data-driven procedures estimating the real cohomology groups of $(X,d,cmu)$. Consider the Hodge-…

Let $\mathsf{pmm}$-spaces be the metric-measure spaces considered in the paper, and let … fails to induce a true metric structure on the space of $\mathsf{pmm}$-spaces. This is mot…

The unbounded Sierpiński carpet and its cable system are metric measure spaces with Hölder exponents $\beta_1-\alpha_1$ and $\beta_2-\alpha_2$ arising from the relevant Hölder regu…