23 problems

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…

Let $d$ be a positive integer, let $(\mathbb{R}^d,g,\mu)$ satisfy the curvature-dimension condition $\operatorname{CD}(\rho,\infty)$ for some $\rho>0$, and let…

A CD metric measure space is a metric measure space satisfying a synthetic curvature-dimension condition of $\mathsf{CD}$ type. Let $f:\overline I\rightarrow [0,\infty)$ be continu…

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 $N\in(1,\infty)$ and let $X$ be a proper, complete, geodesic metric space with a Radon measure $\mathfrak{m}$. Consider the Minkowski $N$-cone $[0,\infty)\times_r^N X$. Minkows…

Generalized-cone characterization conjecture. The space $Y$ satisfies $\textnormal{\textsf{TCD}}(-\kappa N,N+1)$ if and only if $f”-\kappa f\leq0$ in the distributional sense in…

Consider the collapsing family of unitary conformal field theories described above, with $\lambda_{min}$ the minimal eigenvalue of the non-negative operator $L_0+\overline{L}_0$. R…

Curvature-exponent conjecture. Among all sub-Finsler Heisenberg groups, only the sub-Riemannian one has

Sub-Finsler Carnot group curvature-dimension conjecture. The metric measure space $(G,\mathsf d_{SF},\mathfrak m)$ does not satisfy the $\mathsf{CD}(K,N)$ condition for any…

Curvature exponent five conjecture. The metric measure space $(\mathbb{H},\mathsf d,\mathscr{L}^3)$ satisfies $\mathsf{MCP}(0,5)$ if and only if the reference norm is the $\ell^2$-…

Failure of the curvature-dimension condition conjecture. The metric measure space $(G,\mathsf d_{SF},\mathfrak m)$ does not satisfy the $\mathsf{CD}(K,N)$ condition for any…

Let $u\colon\mathbb{R}^n\to\mathbb{R}^m$ be a $1$-Lipschitz map with respect to the Euclidean norms, and let $(\mathbb{R}^n,\lVert\cdot\rVert,\mu)$ be a weighted Riemannian manifol…

Curvature-dimension conjecture. For $b1 geqslant 1$, the $b1$-Grushin plane satisfies $\mathrm{MCP}(K,N)$ if and only if $K\leqslant 0$ and

An expander graph family is a growing family of combinatorial undirected graphs $(G_i)$ of constant degree $D$ such that the first positive eigenvalue of $-\Delta_{G_i}$ is uniform…

Model-density conjecture. The model densities on which $\bar \lambda_{LS}[\mathsf{CD}_{1}(K,N),D]$ is attained are the same as the model densities determining the sharp constants…

Generalization conjecture. The above theorem should generalize to the case $\kappa \geq 0$ for positive $N$.

Let $u\colon\mathbb{R}^n\to\mathbb{R}^m$ be a $1$-Lipschitz map, and let $\nu$ be the measure governing the disintegration into maximal closed convex leaves $\mathcal{S}$ on which…

$CD(0,m)$ characterization conjecture. Then the $CD(0,m)$-condition holds, that is,

A metric measure space $(X,d,m)$ is an Alexandrov space when its metric has curvature bounded below in the sense of Alexandrov. The notation $\operatorname{\mathsf{RCD}}^*(K,2)$ de…

Let a weighted manifold be diffeomorphic to $\mathbb{R}^n$ and satisfy the separate curvature bounds … Here $\operatorname{Ric}_g$ is the Riemannian Ricci tensor and $\nabla_g^2V$…

Let $(\mathbb{R}^n,g,\mu)$ satisfy $CD(\rho,\infty)$ with $\rho>0$, and let $\gamma^n_\rho$ be the $n$-dimensional Gaussian measure. A map pushes forward $\gamma^n_\rho$ onto $\mu$…

Let $(\mathbb{R}^n,g,\mu)$ be a connected weighted manifold satisfying $CD(\rho,\infty)$ with $\rho>0$, and let $\lambda_k(\mathbb{R}^n,g,\mu)$ denote its ordered eigenvalues. Spec…

Let $f535$ be a probability measure on $f535$ that is a $CD(\kappa,N+1)$-needle, where $N\in(0,+\infty]$ and $\kappa\geq0$. Let $\varphi:\mathbb R\to\mathbb R$ be continuous and…