Mathematics Problem Archive

Can every convex polytope in $\mathbb{R}^3$ be partitioned into tetrahedra such that the dual graph has a Hamiltonian path?...

Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors? A light ray is trapped if it ...

How many intersections among an arrangement of pseudosegments in the plane must be added as vertices to allow the pseudosegment arrangment to be exten...

How many polyominoes on $n$ squares are there? A polyomino is a connected interior-disjoint union of axis-aligned unit squares joined edge-to-edge, in...

Is it true that every two sets of $n$ planar points in general position with the same number points on their convex hulls have compatible triangulatio...

For a planar point set $S$, is the number of pointed pseudotriangulations always at least the number of triangulations? A pseudotriangle is a planar p...

Given two sets of numbers, each of size $n$, how quickly can the set of all pairwise sums be sorted? In symbols, given two sets $X$ and $Y$, our goal ...

Consider a polyhedron with simply connected facets (no holes on a facet) and without boundary (every edge is incident to exactly two facets). Can the ...

Can every closed polyhedron be cut along its surface and unfolded into one piece in the plane without overlap? Such an unfolding is called a general u...

Does every simple graph with maximum vertex degree $\Delta \leq 6$ have a 3D orthogonal point-drawing with no more than two bends per edge? A 3D ortho...

What is the complexity of finding a tour of maximum Euclidean length for a planar point set?...

What is the complexity of finding a shortest tour in a solid planar grid graph? A planar grid graph is a graph whose vertices are any set of points on...

What is the complexity of the pallet loading problem? Given two pairs of numbers, $(A,B)$ and $(a,b)$, and a number $n$, decide whether $n$ small rect...

What is the optimal partition of a square into convex pieces such that the circularity of the pieces is optimized? The circularity of a polygon is the...

Given an arbitrary polygon, transform it by a finite sequence of ``vertex-centroid'' moves to a regular polygon. A vertex-centroid move is a translati...

Given a set of $n$ unit-radius balls in $\mathbb{R}^3$, what is the number of lines that are tangent to four of the balls in the set, and miss all the...

Let $C$ be a convex piece of paper; its boundary may be a smooth curve, or a polygon. A perimeter halving folding is a folding of $C$ obtained by iden...

Is there any genus-zero orthogonal polyhedron $P$ built by gluing together cubes face-to-face that cannot be edge-unfolded, where all cube edges on th...

Label the faces of a unit cube with numbers $1$--$6$ as in a die. Place the cube to sit on an integer lattice grid, with one corner at the origin and ...

Let $M$ be a closed polyhedral surface homeomorphic to $S^2$ which is entirely composed of equal regular pentagons. If $M$ is immersed in 3-space, is ...

Partition a given polygon $P$ into $n$ mutually congruent pieces so that the area of $P$ not covered by the union of the pieces is as small as possibl...

Let us say that two rectangles in the place are independent if both their $x$- and $y$-axis projections are disjoint. A set of rectangles is then inde...

Does every convex polyhedron $P$ have a zipper unfolding? A zipper unfolding cuts open $P$ via a single path, necessarily a Hamiltonian path (to span ...

Let $M$ be a compact surface, $I=(-1,1)$, and $f:M\times I\to\mathbb{R}^3$ a differentiable map such that each $f_t$ is an immersion and its induced m...

Is every bubble cluster in $\mathbb{R}^3$ made of spherical pieces meeting according to Plateau's rules necessarily stable, in the sense of having non...

Is a crystal resting on a table under gravity necessarily convex?...

Classify minimizing polyhedral soap-film cones in dimensions five through seven and in all higher dimensions....

Do nonpolyhedral minimizing soap-film cones exist in dimensions four through seven? Construct higher-dimensional examples using triple junctions which...

Classify the singularities allowed in soap films in three-dimensional orbifolds, and more generally in cone manifolds....

If an optimal planar cluster is crossed with a short interval, is the resulting partition optimal in the slab? Is a horizontal mid-height slice optima...

Are there only finitely many combinatorial types of cells in equal-pressure foams in $\mathbb{R}^3$? In particular, can tetrahedra or dodecahedra occu...

If an area-minimizing $k$-dimensional submanifold of $\mathbb{R}^n$ has finite total scalar curvature and $k>n/2$, must it be planar?...

Investigate the relationship between the large-deviation principle, whose rate functions are relative entropies, and dual-connection structures in inf...

Investigate the relationship between the geometry of a conformally flat Riemannian manifold and Matsuzoe's geometric divergence for a conformally-proj...

Characterize the symplectic manifolds with compatible almost complex structure which are locally obtained from a tangent bundle $T(M)$ by combining th...

For a statistical family $f(x;\theta)$, find and study a useful geometry on the product of the sample space and parameter space, accounting for the ch...

Can one remove the “right-angled” assumption in Osajda result?...

Are there torsion-free hyperbolic groups G with cdQ(G)/cdZ(G) < 2/3?...

What can be said about boundaries arising from strict hyperbolization constructions of Charney and Davis, [18]?...

Is there an example of a group G which is hyperbolic relative to some parabolic subgroups that are nilpotent of class ≥ 3 whose Bowditch boundary is h...

Suppose that Z is a compact metrizable topological space, G↷ Z is a convergence action which is topologically transitive, i.e. each G– orbit is dense ...

Find topological restrictions on the ideal boundaries of CAT (−1) cubical complexes....

Is it true that isomorphic Coxeter groups have homeomorphic boundaries?...

Does there exist a Coxeter group Gn with n-dimensional boundary ∂Gn, so that the rational homological dimension of ∂Gn equals 1?...

Under which conditions on the Coxeter diagram of G, the boundary of a Coxeter group is n-connected and locally n-connected?...

Can exotic homology manifolds as in [14] appear as ideal boundaries of Coxeter groups?...

Let $N$ be a closed $n$-manifold and $\Delta$ a flag, no-square triangulation, and let $C(N,\Delta)$ be the associated Davis--Vinberg complex. Is $\pa...

Suppose that ( N1, ∆1) and (N2, ∆2) are closed 3-manifolds equipped with flag-triangulations, so that ∂∞C(N1, ∆1) = ∂∞C(N2, ∆2). Does it follow that ev...

Let Z be a compactum which is a Z-boundary of a group G. Then Z is never a Boltyansky compactum. In the special case when Z is an Markov compactum, so...

Examples of Kleiner and Croke [22], [23] of non-unique boundaries are badly non-locally-connected. Is that essential in having the “flexibility” to hav...