540 problems

Falconer's distance conjecture. If $\dim_{\mathcal H}(E)>d/2$, then $\Delta(E)$ has positive Lebesgue measure.

A Kakeya set in $\mathbf{R}^d$ is a subset containing a unit line segment in every direction. The Hausdorff dimension of a set is its Hausdorff dimension in the ambient Euclidean s…

Fässler–Orponen's conjecture. For $\mathcal{H}^1$-almost every $\theta\in[0,1]$, both

Oberlin's conjecture. Then

A collection of line segments in the plane can be replaced by the associated full lines without increasing its Hausdorff dimension. More generally, let a collection of line segment…

A minimal graph is a hypersurface given by the graph of a function from Euclidean space to the real line, and such graphs are automatically locally area-minimizing. Bernstein's con…

Let $\mathbb S^{n-1}=\{x\in\mathbb R^n:\|x\|=1\}$ be the unit sphere, and let $\alpha_n$ be the supremum of the normalized surface measure of a measurable subset of…

Let $E \subset \mathbb{R}^2$ be an $\alpha$-Furstenberg set, meaning that for each direction $e \in S^1$, there exists a line segment $\ell$ in direction $e$ such that…

A tiling of ${\mathbb R}^3$ into cells of unit volume is being considered, and the surface area of a cell is the quantity to minimize. The proposed minimizing cell is a slightly mo…

Let $\mu$ be an Ahlfors regular measure, and consider its Riesz transform. David–Semmes conjecture. The $L^{2}$-boundedness of the Riesz transform should be sufficient to imply rec…

Vitushkin's conjecture. The set $E$ is non-removable for bounded holomorphic functions if and only if its orthogonal projections have positive length in a set of directions of posi…

Lawson–Osserman conjecture. Every Lipschitz weak solution of the minimal graph system should satisfy the full geometric Euler–Lagrange condition of graph stationarity.

Alexandrov's conjecture. Among all such convex surfaces, the unique surface of maximal area is the doubled disk of radius $\frac{d}{2}$.

Continuous periodic tiling conjecture. The tiling equation $\Omega\oplus X=_{\mathrm{a.e.}}\mathbb{R}^d$ is not aperiodic. Equivalently, if…

Pinned distance conjecture. If

Let $A\subseteq\mathbb{R}$ be an infinite set. Say that $A$ is affinely embedded into a set $E\subseteq\mathbb{R}$ if there are real numbers $a\neq 0$ and $b$ such that…

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

Let $X\subset\mathbb{R}^d$ be a Borel set, with $\operatorname{dim_H}(X)\leq d-1$, that is not contained in a hyperplane. For $y\in X$, let … be the radial projection of $X$ from…

Bukh–Matoušek–Nivasch conjecture. There exists a $k$-flat $\alpha$ in $\mathbb R^d$ (a centerflat) such that

A closed set $E\subset \mathbb{R}^n$ is removable under bounded $K$-quasiregular mappings if every bounded $K$-quasiregular mapping defined on an open set minus $E$ extends to a…

A $(d,k)$ set is a measurable subset of $\mathbb{R}^d$ containing a translate of every $k$-dimensional plane. For $k\geq 2$, a $(d,k)$ Besicovitch set is a $(d,k)$ set of measure z…

Simplex-volume threshold conjecture. The threshold $d-1$ is sharp for guaranteeing that $\mathcal V_d(E)$ has positive Lebesgue measure.

Regularity conjecture. As in the codimension-one case, Almgren's min-max varifolds should have the same regularity as area-minimizing integral currents.

Cheeger's chart-image conjecture. If $(U,\phi)$ is an $n$-dimensional chart, then

Let $P$ be a stationary regular partition of $\mathbb{R}^n$. A partition is strictly area-minimizing in its tangent cones when all tangent cones of $P$ have this property. It is lo…