Mathematics Problem Archive

The McMullen problem on projectively transforming sets of points into convex position...

Opaque forest problem on finding opaque sets for various planar shapes...

Finding matching upper and lower bounds for k-sets and halving lines...

For each arrangement of points in which the rectilinear crossing number is minimized, is the number of halving lines maximized?...

Tripod packing: how many tripods can have their apexes packed into a given cube?...

The Atiyah conjecture on configurations on the invertibility of a certain $n$-by-$n$ matrix depending on $n$ points in $\mathbb{R}^{3}$...

Connelly’s blooming conjecture: Does every net of a convex polyhedron have a blooming?...

Dissection into orthoschemes – is it possible for simplices of every dimension?...

The values of the Hermite constants for dimensions other than 1–8 and 24...

What is the lowest number of faces possible for a holyhedron?...

The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solutio...

Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one...

Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?...

Can every spherical non-convex polyhedron that tiles space by translation have its faces grouped into patches with the same combinatorial structure as...

Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram?...

Is there a general expression for the minimum ropelength of an arbitrary closed knot?...

What constant $1.1 < a \leq 10.76$ governs the lower bound of a closed knot $K$'s minimum ropelength $L(K) \geq a\operatorname{Cr}(K)^{3/4}$?...

Is the upper bound of a closed knot's minimum ropelength linear to its crossing number?...

Is there a general expression for how much the ends of a long rope of radius 1 get closer when a tight open knot is tied into it?...

Does every convex polyhedron have Rupert's property?...

Is there a non-convex polyhedron without self-intersections with more than seven faces, all of which share an edge with each other?...

The Thomson problem – what is the minimum energy configuration of $n$ mutually-repelling particles on a unit sphere?...

Given a unit square, does there exist a point in its plane, inside or outside the square, whose distances from all four vertices are rational? Equival...

Call a finite subset $X\subset S^2$ symmetric if a finite group acts transitively on $X$ by isometries. Determine an explicit universal upper bound fo...

Let $G$ be a planar graph circle-packed in $\mathbb{R}^2$ and let $f:V(G)\to\mathbb{Z}$ change by at most one across every edge. Add from each vertex ...

Among all finite connected vertex-transitive graphs rescaled by their diameters, which one minimizes Gromov–Hausdorff distance to the round sphere $S^...

Let the Euclidean plane or hyperbolic plane have a triangulation whose triangles have diameter at most $r$. Suppose that for every pair of radius-$r$ ...

Characterize the graphs that occur as tangency, or nerve, graphs of sphere packings with disjoint interiors in $\mathbb{R}^d$....

Prove that every sphere packing in $\mathbb{R}^3$ whose tangency graph is $\mathbb{Z}^3$ has at most one accumulation point in the one-point compactif...

If $M^n$ is a closed manifold with a strictly convex projective structure, is it cubulated?...

Suppose a hyperbolic $3$-manifold $M$ admits a CR structure, not necessarily a spherical one. Can the deformation theory of convex real projective str...

If $M$ is a closed hyperbolic $3$-manifold, is every projective structure on $M$ convex?...

Let $N$ be the closed $3$-manifold obtained by gluing two copies of the figure-eight knot complement along their torus boundaries by a homeomorphism. ...

Let $N$ be a closed $3$-manifold whose JSJ decomposition contains only hyperbolic pieces. Does $N$ admit a convex projective structure?...

Geometric transitions are continuous paths of geometries that abruptly change type in the limit. Understand all transitions between the eight Thurston...

The renormalized volume of quasi-Fuchsian groups gives a function $\rho:\mathcal{T}(S)\times\mathcal{T}(S)\to\mathbb{R}$. Is $\rho$ a metric on the Te...

Given $R>0$ and an integer $n\geq4$, is there an $\varepsilon>0$ and a finite set of hyperbolic $n$-simplices such that every closed cone $n$-manifold...

Rivin's experimental results appear extremely regular, possibly indicating additional structure. Investigate this phenomenon....