Mathematics Problem Archive

Do there exist rectangles that may be partitioned into a finite number $n$ of rectangular pieces of equal area but with all perimeters different?...

Let $M$ be a complete Riemannian manifold of even dimension. Suppose that its Euler characteristic $\chi(M)$ and the convergent Gauss--Bonnet curvatur...

Is there a stable cluster of bubbles in $\mathbb{R}^3$ in which some bubble is topologically a torus?...

For $k\leq n+1$ and prescribed volumes in $\mathbb{R}^n$, prove that the standard $k$-bubble is uniquely area-minimizing. For $n=3$, are standard clus...

Let $A_X(v_1,\ldots,v_k)$ be the area of a minimizing $k$-bubble of volumes $v_i$ in a Riemannian manifold $X$. For $X=\mathbb{R}^n$, is $A_X$ strictl...

Are all regions in every area-minimizing bubble cluster in $\mathbb{R}^n$ connected?...

Determine the shapes of minimizing clusters of immiscible fluids when different interfaces have different surface tensions, even for planar double bub...

Determine double crystals in $\mathbb{R}^3$ minimizing orientation-dependent surface energy for a cubic Wulff shape and for general Wulff shapes. Must...

Let $k(n)$ be the maximum number of regions meeting at one point in an area-minimizing partition in $n$ dimensions. Determine the growth of $k(n)$ and...

Is the known list of boundary singularities of soap films in $\mathbb{R}^3$ complete? Classify what happens for singular or immersed boundaries and wh...

Determine the smallest density greater than one of a minimal cone in $\mathbb{R}^n$, with variants imposing area-minimizing or isolated-singularity hy...

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?...