51 problems

Let $\mathcal{C}$ be the set of bivariate copulas, equivalently the set of joint laws of random variables $U,V\sim\mathcal{U}(0,1)$. Define Spearman's rho and Spearman's footrule b…

Asymptotic linear-form conjecture. Under the setting of Theorem InfluenceQuantiles-main,

Let $n\ge1$ and $k\ge2$, and let $K_1,\ldots,K_k\subset\mathbb R^n$ be origin-symmetric convex bodies. Kalantzopoulos–Saroglou's conjecture. If … for all $x_i\in K_i$,…

Let $n\ge1$ and $k\ge2$. Let $\rho:\mathbb R\to\mathbb R_+$ be non-increasing, and let $f_1,\ldots,f_k:\mathbb R^n\to\mathbb R_+$ be even integrable functions. For … assume … for a…

Let $k\ge1$. On a regular problem with an equispaced dense design, consider a degree-$k$ temporal local-polynomial forecaster on the tangent bundle, obtained by geodesic or barycen…

Let $\mathcal{C}:[0,1]\times[0,1]\to\mathbb{R}$ be twice continuously differentiable up to the boundary, symmetric in its two variables, zero on the diagonal, and satisfy…

Let $d$ be a positive integer, let $(\mathbb{R}^d,g,\mu)$ satisfy $\operatorname{CD}(\rho,\infty)$ for some $\rho>0$, and define the centered Gaussian probability measure … where…

Milman's contracting transport conjecture. There exists a map

Let $B$ and $V$ be weight matrices satisfying Assumption, with $V\neq\mathrm{Id}$, and let $(\rho_t^\beta)_{t\geq 0}$ be the token-distribution evolution on ${\mathbb S}^{d-1}$. Le…

Liu–Pego's local convexity conjecture. Every such map $\phi$ must necessarily be locally convex in $\Omega$.

Let $\boldsymbol{A},\boldsymbol{B}\rhd 0$ be positive-definite covariance operators on $\mathcal{H}=\mathbb{R}^{d}$, and let $\pi_{\varepsilon}$ and $\pi_{0}$ denote the entropic a…

Let $W_{2,\mathrm{DPT}}$ be the quantum Wasserstein divergence defined from the De Palma–Trevisan quantum Wasserstein distance by … Here $D(\rho,\sigma)$ is the original quadratic…

Let $\mathsf{P}_{s,\beta}$ denote a one-dimensional non-singular Riesz gas with homogeneity parameter $s$, and let $\mathsf{Leb}_{\mathbb{R}}$ be Lebesgue measure on $\mathbb{R}$.…

Failure of the measure contraction property with strong Goh–Legendre geodesics. If there is a strong-Goh–Legendre geodesic $\gamma$, then $\mathsf{d}^2:M\times M\to\mathbb{R}_+$ fa…

For a separable Hilbert space $\mathcal{H}$, let $\mathbf{C}_p(\mathcal{H})$ denote the set of all possible $p$-cost operators. Monotonicity conjecture for quantum $p$-cost operato…

Generalized barycentric projection conjecture. The map $cmathcal{T}_{cgamma^*}$ is a Monge mapping for the $coperatorname{GW}(cmathbb{S},ctilde{cmathbb{X}}_{cgamma^*})$ problem. Th…

Let $M$ be a closed connected symplectic manifold with symplectic form $\omega$, and endow $M$ with a metric inducing its topology. Let ${\mathcal{S}}(M,\omega)$ be the collection…

De Palma–Trevisan conjecture. Up to non-degeneracy assumptions on the $A_j$ ensuring definiteness, the function $d_{\mathcal{A}}$ is a true metric on the set of states on…

Let $P$ be the probability measure and let $\bar P$ be its Wasserstein barycenter. Let ${\bar P}_n$ denote the empirical Wasserstein barycenter based on $n$ observations, and let…

Gaussian $n^{-1}$ convergence conjecture.

Squaring conjecture. In this setting,

Variance conjecture.

Let $d \ge 2$, let $\mu_i \preceq_c \nu_i$ for $i=1,\ldots,d$, and let $\vec\mu=(\mu_1,\ldots,\mu_d)$. A monotone coupling of $\vec\mu$ is the unique monotone probability measure w…

RCD characterization conjecture. The metric-measure space $(X,d,\mu)$ is $RCD(K,\infty)$ if and only if the five gradients inequality holds.

Bounded-geometry conjecture. The manifold $({\mathcal{M}},g)$ is of bounded geometry if and only if the five gradients inequality holds.