211 problems

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…

Given an analytic family $\mathcal{F}$ of curves in $\mathbb{R}^{2}$, characterize when there exists a Borel set $E\subset\mathbb{R}^{2}$ with Lebesgue measure $|E|=0$ such that, f…

Let $G$ be a compact connected abelian group. Is there a sufficiently small-measure regime in which every compact set $A\subseteq G$ satisfying $\mu_G(A+A)<3\mu_G(A)$ is contained…

Let $(u,K)$ be a generalized global minimizer of the planar Mumford–Shah functional $\mathcal{E}(u,K;B)=\int_{B\setminus K}|\nabla u|^2\,dx+\mathcal{H}^1(K\cap B)$ for every bounde…

Let $\Sigma\subset\mathbb S^3$ be the minimal Möbius band specified in White's conjecture, and let $C(\Sigma)=\{tx:t\geq 0,\ x\in\Sigma\}\subset\mathbb R^4$. The conjecture asserts…

There exist an absolutely continuous probability measure $\mu$ on $\mathbb{R}^2$ and a number $t\in(0,1/4)$ such that, for every pair of perpendicular lines in the plane, the four…

For every integer $n\ge 3$, there exists a constant $C_n>0$ such that, for every nonzero real-valued function $u$ satisfying $\Delta u=0$ in $B_4\subset\mathbb{R}^n$ and $u(0)=0$,…

Yu's conjectures assert, for the relevant pairs of Cantor sets and their radial projections or division sets, that the corresponding sets have nonempty interior. The supplied sourc…

For every pair of positive integers $n,m$, every bounded open set $\Omega\subset\mathbb{R}^n$, and admissible data $g:\Omega\to\mathbb{R}^m$, consider minimizers…

Determine whether, for every dimension $n$ and every algebraic projection multiplicity $Q$, there exists a constant $C(n)$ such that every confined area-minimizing rectifiable curr…

Let $X$ be a Hadamard space and let $r=\operatorname{asrk}(X)$ be its asymptotic rank. For every integer $k\ge r$, there exists a constant $C=C(X,k)>0$ such that every integral $k$…

Let $X=(X_t)_{t\ge 0}$ and $Y=(Y_t)_{t\ge 0}$ be independent Markov processes taking values in a common metric measure space $(E,d,\mu)$. Define the collision-time and collision-po…

Let $K=K(n,m,\Gamma)\subset\mathbb{R}^{2}$ be a Bedford–McMullen carpet with $n>m\ge 2$ and $1<\#\Gamma<nm$. For every isometry $f:\mathbb{R}^{2}\to\mathbb{R}^{2}$, if…

Let $\mathbb{T}$ be a set of length-one $\delta$-tubes in $\mathbb{R}^n$, and let $U(\mathbb{T})$ denote their union. Assume that the tubes have $\delta$-separated directions. Kake…

Falconer's distance set conjecture. If $\dim_{\mathcal{H}}E>\frac{d}{2}$, then

Let $L$ be a set of line segments in $\mathbb{R}^n$. For each line segment $\ell$, let $\widetilde\ell$ denote the line containing $\ell$. Keleti's line segment extension conjectur…

Oberlin's conjecture. Then

Maggi's conjecture. Wulff shapes are the unique such sets.

Let $\gamma:[0,1]\to\mathbb S^2$ be a $C^2$ curve satisfying … For each $\theta$, let $p_{\gamma(\theta)}$ denote projection onto the line in direction $\gamma(\theta)$, and let…

Nonabelian Brunn–Minkowski conjecture. For every pair of compact sets $A,B\subseteq G$,

Let $\mathcal L_{SL_2}$ be the set of lines in $\varmathbb{R}^3$ of the form $\ell_{(a,b,c,d)}=(a,b,0)+\operatorname{span}(c,d,1)$ with $ad-bc=1$, and let $\mathcal L_{SL_2}^*$ als…

Let $E\subset \mathbb{R}^d$ be a Borel set, let $\dim_H(E)>d-1$, and let $s\leq d-1$ be a real number. Radial projection exceptional-set conjecture. … This conjecture strengthens t…

Let $me3$, and let $a_1,\dots,a_m>0$ satisfy … Let $\Omega_1,\dots,\Omega_m\subseteq\mathbb{R}^{n+1}$ be a minimizing partition for the Gaussian multi-bubble problem: the sets cover…

Finiteness conjecture. The space-time singular set has only finitely many components.

An open surface is a non-compact surface without boundary. Let $S$ be an open, connected, orientable surface. Ros's conjecture. Every such surface can be properly and minimally emb…