Mathematics Problem Archive

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 ...

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...

Suppose that G is a CAT (0) group which does not split over a small subgroup. Does it follow that ∂∞G is unique?...

Is the boundary well-defined for groups acting geometrically on CAT (0)-cube complexes? More precisely, suppose that X1, X2 are cube complexes which ad...

What topological invariants distinguish boundaries? In particular, what topological properties of boundaries are quasi-isometry invariants? Does somet...

If G acts geometrically on two CAT(0) spaces, are the resulting boundaries cell-like equivalent? (That is, does there exist a space Z with cell-like m...

Is there a convex core for the diagonal action of G on X1 × X2? (A special case is surface groups G with X1 and X2 corresponding to different hyperboli...

Define a topology on the set of quasi geodesics in a (proper geodesic, or coarsely homogeneous, or cocompact) CAT (0) space which (1) has a description...

Any CAT (0) group has no infinite-torsion subgroups. A Euclidean retract is a compact space that embeds into some Rn as a retract. A compact metrizable...

Let G be a hyperbolic group and ∂EZ G be its EZ boundary. Is it true that ∂EZ G is equivariantly homeomorphic to the Gromov boundary of G?...

Can there be two different boundaries in the sense of Z-structures for a group G that are not cell-like equivalent?...

Is the property of splitting over a 2-ended subgroup an invariant of Bestvina boundaries? Some necessary conditions are known for compact, metrizable ...

Do Papasoglu's three results for finitely presented one-ended groups extend to all finitely generated groups: quasi-isometry invariance of the JSJ dec...

Are splittings overZ2 (orZn) invariant under quasiisometry? The analogous problem also makes sense for the JSJ decompositions....

Suppose G is finitely generated and there is a sequence of quasicircles that separate its Cayley graph. Is G virtually a surface group?...

If G is finitely generated with asymptotic dimension ≥ n, and X is a subset of the Cayley graph with asymptotic dimension ≤ n − 2 that coarsely separat...

Do all homogeneous continua (with dimension greater than 2) have this property?...