97 problems

For every integer $n\ge 2$, every complete simply connected Riemannian $n$-manifold $(M^n,g)$ with sectional curvature $\operatorname{Sec}_g\le \bar k\le 0$, and every volume $V>0$…

Suppose that $(M,g)$ is a compact perturbation of Schwarzschild-anti-de Sitter of mass $\boldsymbol{m}\text{?}$ and has scalar curvature $R_g\geq -6$. If $m_{\partial M}\geq 0$ sat…

Let $Q=(0,1)^3$. For every measurable set $E\subset Q$ of finite perimeter and volume $|E|=v\in(0,1/2]$, the relative perimeter $P(E;Q)$ satisfies … Moreover, equality cases, up to…

Let $S$ be a horizontal strip in $\mathbb{R}^2$, and let the density be $\rho(x)=1$ on $S$ and $\rho(x)=\lambda>1$ on $\mathbb{R}^2\setminus S$. For every $\lambda>1$ and every adm…

For $n\ge 2$, let $K\subset\mathbb{R}^n$ be a centrally symmetric convex body, meaning $K=-K$, and let $B\subset\mathbb{R}^n$ be a Euclidean ball satisfying…

Viterbo's conjecture.

Let $n\geqslant 2$, and let $M^n(c)$ be an $n$-dimensional model space of constant sectional curvature $c\in\mathbb{R}$. Let $K\subset M^n(c)$ be a $\lambda$-convex body, and let…

Let $\mathbb{R}^{n}$ be equipped with a smooth, radial, log-convex density, and consider regions of prescribed volume. Brakke's conjecture. Balls centered at the origin are isoperi…

KLS conjecture. The KLS constant satisfies

Let $q$ satisfy $2\leq q\leq n+2$, and let $\Omega$ be a $q$-cluster on $\mathbb{S}^n$ with prescribed volume $V(\Omega)=v\in\operatorname{int}\Delta^{(q-1)}_{1}$. A standard bubbl…

Let $S_n$ be the symmetric group, and let $T_n$ be its transposition graph. For $\mathcal{A} \subset S_n$, write $\partial \mathcal{A}$ for its edge-boundary in $T_n$. Let…

Hadamard-type conjecture. (i) $\mathcal G$ is concave if and only if $G_v\geq 0$ for every $x\in\Omega$. (ii) $v$ is $-\infty$-concave, meaning that $\mathcal G$ has convex level s…

Optimal-scale conjecture. For each $C>0$ there is an $\varepsilon>0$ such that, for every monotone ${\cal F}$, there exists

Let $G$ be a finite, connected, vertex-transitive graph. Write $\operatorname{diam}(G)$ for its diameter and, for a vertex set $S$, let $\partial S$ denote its edge boundary. Isope…

Let $G$ be a finite graph and define … Here $\partial S$ is the vertex boundary used in the paper, and $p^G$ is the percolation threshold. Isoperimetric-threshold conjecture. If…

Let $H_{\alpha,\Gamma}$ be the planar Hamiltonian with attractive interaction of strength $\alpha>0$ supported by a loop $\Gamma$ of fixed length $L$, and let…

Let $K\subset\mathbb{R}^n$ be a convex body, and let $\Lambda_{\mathrm{FK}}(K)$ be its normalized minimal first Dirichlet eigenvalue. Schmuckenschläger's reverse Faber–Krahn bound.…

Let $N\ge2$, let $\Omega\subset\mathbb R^N$ be a bounded domain, and let $\nu=(N-2)/2$. Denote the Dirichlet eigenvalues of $\Omega$ by…

Let $K,Q e othing$ be convex bodies in $\mathbb{R}^n$, and let $c(K\times Q)$ denote the symplectic capacity of their Lagrangian product. Viterbo's isocapacitary conjecture. The La…

Let $\Sigma$ be a compact surface in an asymptotically flat manifold with nonnegative scalar curvature $R\geq0$. Let $M$ be the mass and define the Schwarzschild radius by $r_H=2M$…

Let $\mu_{1}$ and $\mu_{2}$ be log-concave probability measures on Riemannian manifolds $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$, respectively. For $a$ in the domain of the isoperim…

Let $(\Sigma,g)$ be a compact Riemannian surface with boundary that is not diffeomorphic to the disk. Let $\sigma_1(\Sigma,[g])$ denote its first normalized Steklov eigenvalue, and…

Let $\mathbb{D}$ be the disk, let $h_t^+$ denote the corresponding Steklov data for the critical ellipse, let $E_t=\{x^2+t y^2=1\}$, and let $co(E_t)=\{x^2+t y^2\leq 1\}$. Equip…

Let $K\subseteq\mathbb R^m$ be an origin-symmetric convex body, meaning that $x\in K$ if and only if $-x\in K$. Let $\mathsf{SL}_m(\mathbb R)$ denote the group of volume-preserving…

Let $K ot i 0$ be an origin-symmetric convex body in $\mathbb R^m$, meaning that $x\in K$ if and only if $-x\in K$. Write $\lambda(K)$ for the first Dirichlet eigenvalue of $K$, an…