72 problems

Let $(X,d,\mathfrak{m})$ be an $RCD^{\ast}(0,N)$ space, where $RCD^{\ast}(0,N)$ denotes the relevant synthetic non-negative Ricci-curvature condition. Milnor's conjecture. The fund…

Let $(X,d,\mathfrak{m})$ be an $RCD^{\ast}(K,N;D)$ space. If its fundamental group $\pi_1(X)$ is finite, then Fukaya–Yamaguchi conjecture. It contains an abelian subgroup $A$ of in…

Let $(M^3,g)$ be a complete, non-compact Riemannian manifold. Say that $(M^3,g)$ is Ricci pinched when it satisfies the Ricci-pinching condition used in Hamilton's conjecture. Hami…

Volume growth and asymptotic cone dimension conjecture. (1) If

Ricci lower-bound conjecture. Then

Let $(M,\xi)$ be a contact 3-manifold equipped with a compatible metric $g$. Write $Ricci(g)>0$ for positivity of its Ricci curvature, and let $(\mathbb{S}^3,\xi_{std})$ denote the…

Let $M$ be complete, let $O$ be a fixed origin, and suppose that for some $r_0>0$ the Ricci curvature is bounded below by $(n-1)/(4r^2)$ whenever $r\geq r_0$. Higher-dimensional Ri…

Let $(M,g)$ be a complete, noncompact manifold with only one end: outside a compact set it is diffeomorphic to a spherical shell $S^{n-1}\times\mathbb{R}$. Let $g_0$ denote the Euc…

Let $(M^n,g)$ be a complete Riemannian manifold with smooth nonempty boundary $\partial M$. Suppose $M$ has nonnegative Ricci curvature and $\partial M$ is uniformly convex, meanin…

Milman's contracting transport conjecture. There exists a map

Let $d$ be a positive integer, let $(\mathbb{S}^d,g)$ be a Riemannian sphere, and let $\operatorname{vol}_g$ denote its volume measure. Suppose that … for some $\rho>0$. Let…

Let $(M,h)$ be an $n$-dimensional Riemannian manifold, and let $g$ be a Lorentzian metric such that $(M,g)$ is a globally hyperbolic spacetime. Write…

Unboundedness conjecture. Under the assumptions above, the quantity

Let a bitnet be a binary Bayesian network equipped with its Fisher information metric, and let $\langle R\rangle$ denote its volume-averaged Ricci scalar. Rodríguez's conjecture. F…

Let $(M^3,g)$ be a complete non-compact Riemannian manifold, let $q\in M$, and suppose that $Rc\geq 0$ and $M$ has maximum volume growth. Writing $R$ for the scalar curvature and…

Let $V_\ell$ denote the eigenspace indexed by $\ell$ in the Zeitlin model on $SU(N)$, and let $\widetilde{r}_\ell(N)$ be the averaged Ricci curvature in that eigenspace. Eventual n…

For fixed $N$, let … let $\lambda_i=i(i+1)$, and let $H_j=1+1/2+\cdots+1/j$ be the $j^{\text{th}}$ harmonic number. The conjectured Wigner $6j$ identities. For all…

For given $m \in \mathop{\mathrm{{\Bbb N}}}\nolimits$, $\kappa \in \mathop{\mathrm{{\Bbb R}}}\nolimits$ and $v,D>0$, consider connected oriented closed Riemannian manifolds…

Fix real numbers $C$, $a$, and $b$. Consider quantum Riemannian $1$-spaces with measure whose associated Hilbert space comes from a unital involutive algebra and a state, with non-…

Let $(X,\omega_X)$ be an $n$-dimensional compact normal Kähler space with smooth Kähler metric $\omega_X$, and let $\omega=\omega_X+\sqrt{-1}\partial\bar\partial u$ be a singular K…

Ollivier-Ricci curvature and LSI. There exists a universal constant $c<\infty$ such that whenever the Ollivier-Ricci curvature $\tilde{\kappa}$ is positive, we have

RCD versus smooth curvature conjecture. Such an $n$ and $M^n$ exist.

RCD no-$\mathbb{RP}^2$ cone conjecture. Then $Y\approx S^2$ or $Y\approx \overline{D}^2$. If the $X_i$ have empty boundaries, only the first possibility can occur.

Brue–Pigati–Semola conjecture. $(X,\mathsf{d})$ is locally biHölder homeomorphic to a smooth, complete Riemannian manifold $(M^3,g)$.

Let $M$ be a manifold. Suppose $M$ admits a Riemannian metric with nonpositive sectional curvature everywhere and negative definite Ricci curvature at some point. Connell–Wang's co…