93 problems

Viterbo's conjecture.

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 $\mathbb{R}^{n}$ 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…

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 $c(Ktimes Q)$ denote the symplectic capacity of their Lagrangian product. Viterbo's isocapacitary conjecture. The Lagr…

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…

KLS conjecture. The KLS constant satisfies

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 $xin K$ if and only if $-xin K$. Write $\lambda(K)$ for the first Dirichlet eigenvalue of $K$, and le…

Let $\Omega$ be a domain of finite measure, let $\mathbb B$ denote the unit ball in $\mathbb R^d$, and let $\omega_d=|\mathbb B|$. For $m,t\in\mathbb N$, write…

Borisenko's conjecture. Among all such sets, the sets maximizing surface area are precisely the lenses of volume $V$.

Let $utext{?}$ and $u$? No. Let $\beta$?

Let $M\subset\mathbb S^{n+1}_1$ be a spacelike, compact, star-shaped, and $k$-convex hypersurface. For $1\leq k\leq n$, let $\mathcal A_k$ denote the quermassintegral-type quantity…

Let $\Sigma\subset\mathbb{R}^{n+m}$ be an $n$-dimensional smooth minimal submanifold with smooth boundary $\partial\Sigma$. Write $|\Sigma|$ for its $n$-dimensional volume,…

Let ${\mathbb B}_n$ be the unit ball in $\mathbb C^n$, with Lebesgue measure $dv(z)$, and define the hyperbolic measure by … For $0<p<\infty$ and $\alpha>n$, let $A^p_\alpha$ be th…

Let $(M^m,g)$ be a compact manifold with smooth boundary $\partial M$, and let $\eta\in\Omega^1(M)$. Let $\mathfrak{B}_M$ denote the relevant gauge-trivial subspace, and for an adm…