21 problems

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

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 $X$ 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…

Let $(X,d,m,\mathcal E,\mathcal F)$ be an MMD space, and let $d_m$ denote the martingale dimension of its associated diffusion. Suppose that $(X,d,m,\mathcal E,\mathcal F)$ satisfi…

Let $G=({\mathcal X},E)$ be a graph endowed with the counting measure. For $z\in {\mathcal X}$, let $S_2(z)$ denote the vertices at graph distance two from $z$, and for…

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…

Let $(M_1,d_1,{\Greekmath 0116}_1)$ and $(M_2,d_2,{\Greekmath 0116}_2)$ be metric measure spaces admitting heat kernels satisfying the heat kernel estimate (hk), with the same expo…

Let $n\geq 1$ be an integer, and let $\theta,\theta'\in\mathbb{S}^n$. The map $\Phi_{\mathbb{S}^n}$ is defined through the classical multidimensional scaling construction, and…

Volume-density conjecture. We have

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 $B^H$ be a fractional Brownian motion with Hurst index $H\in(0,1)$, and let $\varphi(W)$ be the Hunt process on $\operatorname{gr}(B^H)$ induced by a Wiener process through the…

Curve Histogram Conjecture. The distance histogram $H_{\mathcal{X}}$ determines a fully regular plane curve up to isometry.

Let $c4$ and $c4'$ be finitely isomorphic ergodic filtrations. Assume that both filtrations are nonstandard and that both have virtual matrix distributions for the sequences of ass…

Finite-fiber submetry conjecture. Under these assumptions,

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 $1 < p \in \mathbb{R}$ and let $X$ be a metric measure space. Let $\Delta_p(X)$ denote the $p$-harmonic boundary of $X$, and let $BD^p(X)$ and $HBD^p(X)$ denote the bounded Roy…

Let $(\boldsymbol{X}_\bullet,\boldsymbol{X}'_\bullet)$ be a geodesic hinge with endpoint distances $R$ and $R'$, and let ${\sf f}_0,{\sf f}_1,{\sf f}'_1$ be the associated function…

Let $\mathcal V^\Lambda=(\mathcal V_t^\Lambda)_{t\geq 0}$ be the tree-valued $\Lambda$-resampling dynamics. Atomlessness conjecture. For Lebesgue almost every $t$, the metric measu…