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...

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 ...

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...

Prove that the unit-volume polyhedron with shortest total edge length is an equilateral triangular prism, and establish existence of a minimizer....

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...

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...

Given a statistical manifold, determine conditions for the existence of a family of probability distributions whose induced statistical manifold coinc...

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

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 $...

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 ...

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?...

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

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

Extend the known necessary conditions for a compact metrizable space $X$ to be the boundary of a proper cocompact CAT(0) space, or give a complete cla...

Does every CAT(0) group have finite asymptotic dimension?...

Develop a theory for analysis on the ideal boundaries of relatively hyperbolic groups, as it is done for hyperbolic groups....

In what generality does quasiconformal imply quasisymmetric?...

Are all quasi-isometric embeddings between higher-rank symmetric spaces either isometries or algebraic in this way?...

[Cannon–Thurston] Is this action conjugate to a conformal action?...

Let H ⊂ G be a hyperbolic subgroup of a hyperbolic group (we do not assume that H is quasiconvex). Is it true that there exists an equivariant continu...

What is the Poisson boundary of the free group with an arbitrary measure?...

Characterize the hitting measure ν on PMF obtained from the random walk by mapping classes on Teichm¨ uller space. Is it absolutely continuous with re...

What is the Poisson boundary of Outer space?...

Is there are meaningful structure theory for lacunary hyperbolic groups? Can one define a useful boundary for such groups? Is it true that either Out(G...

Every relatively hyperbolic group has cut points in all of its asymptotic cones. To what extent does the converse hold? Characterize the finitely gene...

For the fundamental group G of a closed hyperbolic n-manifold consider a short exact sequence 1 →Zp → Γ → G → 1. Is the group Γ residually finite? In o...

Let G be a Gromov-hyperbolic Coxeter group. Does G admit a discrete embedding in Isom(Hn) for large n?...

Let G ⊂ Isom(Hn) be a discrete torsion-free finitelygenerated subgroup without abelian subgroups of rank ≥ 2. Is it true that (a) cdZ(G) ≤ Hdim(Λc(G)) ...

Generalize Bestvina-Feighn combination theorem from graphs of groups to complexes of groups....

There is a theory of quasi-convex groups acting on Gromov hyperbolic spaces, generalizing the theory of convex-compact groups of isometries of the rea...

Extend this relation of Anosov structure and dynamics on the limit set to representations of other hyperbolic groups....

Obtain new rigidity results for embeddings of realhyperbolic lattices into higher-rank semisimple Lie groups in terms of the boundary maps....

Consider Teichm¨ uller spaceT (S) with Teichm¨ uller metric. Does it have quadratic isoperimetric inequality?...

Compute cogrowth for notable subgroups $H\subset G$, where cogrowth is the growth of the Schreier graph $\Gamma_{G/H}$. In particular: prove that the ...