198 problems

Least-perimeter six-region partition conjecture. The least-perimeter way of dividing the unit disk into six regions of prescribed areas is given by the configuration shown in Figur…

Let $V$ be a stationary integer rectifiable varifold, let $x_0$ be a point with a unique flat tangent cone, let $\Theta(V,x_0)$ denote its density at $x_0$, and let…

Let $[0,1]^n$ be the unit cube in $\mathbb{R}^n$, and let a cubical subset be a compact subset of $[0,1]^n$ whose boundary is contained in finitely many hyperplanes with outer norm…

Let $[0,1]^n$ be the unit cube in $\mathbb{R}^n$ with the Euclidean metric. For a measurable set $U\subset [0,1]^n$ of locally finite perimeter, let $\operatorname{Vol}(U)$ denote…

A circle packing is a packing of congruent circles in the plane, and a packing is diffusively dominant if its mass distribution diffusively dominates that of every other packing. C…

Let $A_1$, $A_2$, and $A_3$ be positive real numbers. Consider planar graphs enclosing and separating finite areas $B_1$, $B_2$, and $B_3$ satisfying … A standard triple bubble is…

Let $H^+\subseteq\mathbb{R}^{(a,b)}$ be the positive pseudosphere and let $N^+\subset H^+$ be a neck, with Riemannian volume $\operatorname{Vol}N^+$. Flat-neck volume conjecture. T…

$(n,k)$-set dimension conjecture. If $E\subset\mathbb{R}^n$ is an $(n,k)$ set, then

The continuity-based criterion. Under these assumptions, $A$ does not have property (iv). The conjecture proposes that the conclusion of Theorem 2 remains valid when the relevant g…

Let $X$ be a manifold, and for each open subset $U\subset X$ let ${\cal S}_i(U)$ be the space of continuous valuations on $U$. These spaces form a presheaf ${\cal S}_{i,X}$ on $X$.…

Let $M$ be a properly immersed minimal surface in ${\mathbb R}^3$ with quadratic area growth, meaning that its area growth constant $A(M)$ is finite. A limit tangent cone at infini…

Let $A_0\geq A_2\geq A_1$ be areas, and let a flat torus have area $A_0+A_1+A_2$. A non-tiling minimizing double bubble is a perimeter-minimizing double bubble enclosing areas…

Let a minimizing graph be a least-perimeter partition of the unit disk into regions of prescribed areas. Connectedness conjecture. A minimizing graph separates the disk into connec…

Small-volume triple-bubble conjecture. The minimizers will look like the double bubbles of Figure 2 with a small ball attached, and the phase diagram will look just like Figure 3.

Complete-classification conjecture. The double bubbles of Figure 1 together with the Hexagonal Honeycomb of Figure 4 comprise the complete set of area-minimizing double bubbles for…

No periodic three-volume foam conjecture. There are no least-area divisions of $T^3$ into three volumes that lift to a foam in $\mathbb{R}^3$.

Delaunay-chain transition conjecture. The first phase transition as small equal, or close to equal, volumes grow is from the standard double bubble to a chain of two bubbles bounde…

Slab Lens conjecture. For one very small volume and two moderate volumes, the Slab Lens is optimal.

Small-volume standard double-bubble conjecture. For two small volumes the standard double bubble is optimal.

Connectedness conjecture. An area-minimizing double bubble in $T^3$ has connected regions and complement.

Fix $n\geq 2$ and $0<\delta\ll 1$. Let $\Omega$ be a maximal $\delta$-separated subset of directions on $S^{n-1}$, and for each $\omega\in\Omega$ let $T_\omega$ be a…

David–Mayboroda conjecture. Under these hypotheses, the Weak Geometric Lemma implies the Bilateral Weak Geometric Lemma, and consequently uniform rectifiability of $E$.

Let $M$ be a Riemannian manifold and let a homologically area-minimizing cycle in $M$ have a singular set. White's generic regularity conjecture. Singularities in a homologically a…

Fix a Riemannian $n$-manifold $(M,g)$, $n \geq 2$, and let $U \subset\subset V \subset\subset M$ be open. An integral stationary geodesic network is a geodesic network with integer…

Let $G=(\mathcal{V},\mathcal{E})$ be a simplicial complex. A triangle edge choice function assigns to each triangle of $G$ one of its edges, and it is injective when distinct trian…