Mathematics Problem Archive

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?...

Are diffeomorphisms Rn →Rn dense in the space of all quasiconformal maps?...

Let f: Bn → Bn be a quasiconformal homeomorphism. Can f be approximated by globally quasiconformal diffeomorphisms fj: Bn → Bn? Can this be done so tha...

Find good classes of spaces such that the infinitesimal metric condition (for quasiconformality) implies the local condition. (This is generally true i...

Outside of the boundaries of Fuchsian buildings, what boundaries have the Loewner property?...

Let X be a non-smoothable closed simply connected 4-manifold. Does it admit an Ahlfors 4-regular linearly locally contractible metric? This is wide op...

Take your favorite metric fractal. Is it quasisymmetrically cohopfian? What about the boundaries of hyperbolic groups?...

If ∂∞G is Loewner, then it is quasisymmetrically cohopfian. (Boundaries of Fuchsian buildings provide a good test case for this conjecture.)...

If G is a hyperbolic group and ∂∞G is connected with no local cut points, is there a natural measure class which is quasisymmetrically invariant?...

Can you do analysis on CAT(0) boundaries? With no natural metric, is there any structure beyond topology?...

G = Isom( X) acts on ∂∞(X). Is this action “nice” with respect to the metrics in the previous remark?...