57 problems

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…

Villani's conjecture. If the squared-distance cost satisfies the MTW condition, then all injectivity domains of $M$ are convex.

Let $\Pi$ be the set of all bivariate copulas. For $C\in\Pi$, define Spearman's rho and Gini's gamma by $\rho(C)=12\int_0^1\int_0^1 C(u,v)\,du\,dv-3$ and…

For every metric-measure space $(X,d,\mathfrak m)$, every $K\in\mathbb R$, and every $1<N<\infty$, is $\mathrm{BCD}(K,N)$ equivalent to $\mathrm{RCD}(K,N)$? Equivalently, does the…

Let $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ be the flat unit torus, let $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_n$ be independent samples of $n$ independent points uniformly distribute…

Let $\lambda$ denote the uniform probability measure on $[0,1]$, let $\Pi(\lambda,\lambda)$ be the set of Borel probability measures on $[0,1]^2$ whose two marginals are $\lambda$,…

For every $L\ge 0$, there exists a finite constant $C(L)$, independent of the dimension $d$, such that for every $d\ge 1$ and every globally $L$-Lipschitz function…

Let $(X,Y)$ be jointly Gaussian with $X,Y\sim\mathcal{N}(0,1)$ and correlation coefficient $\rho\in(0,1)$. The exact common information of $(X,Y)$ is conjectured to satisfy…

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…

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…

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

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 $k\ge1$. 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 $W_{2,\mathrm{DPT}}$ 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…