Mathematics Problem Archive

Do least-area partitions of $\mathbb{R}^n$ into regions of unit volume exist? Give the correct definition and determine their regularity....

Is the Weaire--Phelan A15 foam the optimal partition of three-space into equal volumes, and can the existence of a foam in the A15 pattern be proved?...

Prove or disprove successively that the Kelvin cell is the least-area fundamental domain for the BCC torus, that it minimizes among all unit-volume fl...

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...

Prove or disprove that no two-dimensional froth of average bubble area one has shear modulus in any direction exceeding that of the regular hexagonal ...

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

Find the best constant $C$ in the inequality $A\leq C(\int_B H+L)^2$ for a bubble cluster of area $A$, mean curvature $H$, and boundary length $L$. De...

Extend Gromov's Knothe-based proof of the isoperimetric inequality to multiple regions, and determine whether its vector field can be chosen canonical...

Is the cone over the regular tetrahedron the smallest soap film having the entire tetrahedral frame as boundary? For a frame of two rectangles sharing...

For a regular octahedral wire frame, determine the least-area soap film separating the eight regions, both with ordinary and fractional-density soap f...

If a convex planar curve $\Gamma$ has length less than $2\pi$, does it bound a graph of constant mean curvature one? Determine the volume threshold $V...

Is every embedded constant-mean-curvature surface, or every immersed constant-mean-curvature disk, with boundary a round circle necessarily a spherica...

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

Find the best constant $C$ such that the principal curvatures of an area-minimizing hypersurface in low dimensions are bounded by $C/r$ at a point who...

For a statistical manifold $(M,g,\nabla,\nabla^*)$, find conditions under which it can be realized as an $n$-dimensional submanifold of an $m$-dimensi...

The smooth positive probability densities on $S^1$ are diffeomorphic to the manifold of smooth equiaffine transformations of $S^1$. Find the correspon...

Given a Riemannian manifold $(M,g)$, can one always introduce a symmetric $(0,3)$-tensor $T$ so that $(M,g,\nabla,\nabla^*)$ is dually flat? If the co...

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

Determine the fundamental group of a compact affine flat manifold....

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

Do affine minimal surfaces admit isothermal coordinates with respect to their affine fundamental form, analogously to minimal surfaces in Euclidean th...

Let $K(X,Y,Z,W)=h(R(X,Y)Z,W)$ be the curvature $(0,4)$-tensor of a statistical manifold. Investigate the structure of all $(0,4)$-tensors satisfying $...

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

On a statistical manifold, let $D$ be the rank-$(n-1)$ distribution orthogonal to the velocity of the $\nabla$-geodesic from a fixed point. If every s...

Estimate the minimum number of inflection points of a closed curve on a Riemann surface with a projectively flat connection....

For a closed curve on a Riemann surface with projectively flat connection, estimate the number of affine vertices and, assuming isolated inflection po...

Prove existence of a time-global solution to the affine-plane curve evolution equation $\partial x/\partial u=(1+kp)x''$, where $u$ is time, $x''$ is ...

If $(M,g)$ is a complete Hessian manifold, is its tangent bundle with the natural complex structure a Stein manifold? In particular, if a convex domai...

Let $M$ be a compact Hessian manifold and deform its affine structure. Does every sufficiently small deformation admit a Hessian metric?...

Construct compact Hessian manifolds other than direct products of hyperbolic and flat compact Hessian manifolds; in particular, construct one with no ...

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...

What spaces can arise as boundaries of hyperbolic groups? As a sub-problem: For which k do k-dimensional stable Menger spaces appear as boundaries?...

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

What 2-dimensional spaces arise as boundaries of hyperbolic groups?...

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?...

Find further universality phenomena: classes of groups or spaces of different nature whose ideal boundaries are nevertheless all homeomorphic, beyond ...

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...