Mathematics Problem Archive

Does every three-dimensional convex polytope have a non-self-intersecting edge unfolding? Does a minimum spanning tree of the dual edge graph, with a ...

On a centrally symmetric convex body, must every pair of points at maximum intrinsic surface distance be antipodal? Resolve this even for rectangular ...

Determine the minimum number of $d$-simplices needed to triangulate the $d$-dimensional cube, and its asymptotic growth with $d$....

Does the hyperbolic plane admit a smooth isometric immersion into $\mathbb R^4$? More generally, determine the least Euclidean dimension for such an i...

Are the Hermite constants associated with densest lattice sphere packings always rational? Determine their arithmetic nature in dimensions where the e...

What is the largest possible ratio between the sum of edge lengths of a tetrahedron that can pass through or fit inside another tetrahedron and the su...

Does there exist a nondegenerate triangle whose side lengths, three medians, three altitudes, and area are all rational?...

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

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

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

Let $\bm{x}\in\{0,1\}^{\mathbb{Z}}$ be a pattern-Sturmian sequence and define $[H\psi](m)=\psi(m+1)+\psi(m-1)+\lambda x_m\psi(m)$ on $\ell^2(\mathbb{Z...

For the graph $(V,E)$ of a Penrose tiling, define $H$ on $\ell^2(V)$ by $[H\psi](v)=\sum_{w:(v,w)\in E}(\psi(w)-\psi(v))$. Determine the spectrum $\si...

A one--dimensional, unimodular Pisot inflation tiling has pure point spectrum if its first rational \v{C}ech cohomology group has rank equal to the al...

The coincidence rank of a one--dimensional Pisot inflation tiling must divide the algebraic norm of $\lambda$....

Let $1,\alpha,\beta$ be $\mathbb{Q}$-linearly independent, let $Y(\alpha,\beta)$ be their canonical cut-and-project set, and let $\xi_{(\alpha,\beta)}...

Let $1,\alpha,\beta$ be $\mathbb{Q}$-linearly independent, let $Y(\alpha,\beta)$ be their canonical cut-and-project set, and let $\xi_{(\alpha,\beta)}...

For $1,\alpha,\beta$ linearly independent over $\mathbb{Q}$, does $\liminf_{n\to\infty}n\|n\alpha\|\|n\beta\|=0$ imply that the number of distinct pat...

Let $\mathcal{D}\subset\mathbb{R}^2$ be a Delone set BL to $\mathbb{Z}^2$. Does there exist a bi--Lipschitz map $L:\mathbb{R}^2\mapsto\mathbb{R}^2$ su...

Find matching rules in dimension two or three satisfying both: (A) every tile-type discrepancy in a finite patch is bounded by a constant times the bo...

Find matching rules in dimension two satisfying condition (A): for every tile type $\mathfrak t$ and finite region $\mathcal R$, the discrepancy $|N_{...

Determine all $SL_d(\mathbb{R})$--invariant Borel probability measures on $\mathbf{Cl}(\mathbb{R}^d)$ and similarly for the $ASL_d(\mathbb{R})$ action...

Let $E\subset\mathbb{R}^k$ be a totally irrational subspace of dimension $d\ge 1$, and let $Y$ be a cut--and--project set obtained from $E$ using a bo...

An optimist would expect similar inequalities distg(∂−,∂+) <δ =δ(Y ) <∞ (ideally withδ = 2π dim(Y )+1) for metricsg on Y ×[−1,+1]with Sc(g) ≥n(n−1) fo...

[∗] no closed aspherical15 manifold admits a metric withSc > 0....

Problem. Evaluate σ○(X0) and σ◻(X0) for "simple" Riemannian manifolds X0 = (X,g 0). ###◻Dirac operators, because they are invariant under isometries, ...

Conjecture. Bounds on Width and on the Macroscopic Dimension. Complete n-dimensional Riemannian manifoldsX with the scalar curvaturesSc(X) ≥σ > 0 sati...

Conjecture. Bound on the Filling Radius forSc ≥σ > 0., fil.rad[X]≤constn⋅( inf x∈X Sc(X)(x))−2. This, in view ofA,B,C from the previous section, yield...

Conjecture. Asphericity⇒K-Area =∞. The universal coverings ˜X of compact aspherical manifoldsX satisfy K-area( ˜X) =∞. Notice that this inequality, ev...

Conjecture Area Extremality and Rigidity of Symmetric and Einstein Spaces. All Riemannin manifolds with positive and parallel Ricci tensor, in particu...

Spin Problem. All of the above only applies to spin maps f ∶X→X, for which the required twisted Dirac operator defined, and, as on similar occasions we...

Conjecture. All of the above is satisfied, modulo constants, for alln-manifolds, possibly incomplete and/or with boundaries, withSc(X) ≥σ > 0. Namely m...

Conjecture. Sharp Spherical Length Comparison Inequality. Spheres with finitely many punctures are length extremal. In fact – this is, probably equival...

Let $\widetilde X$ be the universal cover of a Riemannian $n$-manifold $X$ homeomorphic to the $n$-torus. Conjecture that $\widetilde X$ has non-posit...

Conjecture Let a domainY ⊂Rn havemean.curv(∂Y ) ≥ n−k+ε for someε> 0 and k = 2,...,n −1. ThenY−1 admits a continuous map onto a(k−1)-dimensional polyh...

Conjecture. C0-ContinuousGuth-GerochLowerVolume Bound for Balls in the Coverings of Essential Manifolds. [57]. The universal coverings ˜X of Q-essenti...

Conjecture. C0-Density of C0-metrics with Volumically Positive Scalar Curvatures. Continuous Riemannian metrics withScvoln > 0 on anX are dense in the...

but there are no apparent examples (if any) where these inequalities are strict. Everything we know aboutK-area+ easily extends to the the FredholmKar...

Conjecture. Hyperbolic Volume Inequality.Then every continuous mapf0∶X→X0 is homotopic to a mapf, such that voln(f(X)) ≤vol(X) where, moreover, this i...

Prove that there is a dimension-dependent constant $c_n$ such that every compact Riemannian $n$-manifold $X$ with $\operatorname{Sc}(X)\geq-\sigma^2$ ...

Given a graph G. Does there exist a Cayley algebra system (or several systems) describing the union of the codimension 1 tensegrity strata in the plan...

Does there exist an embedded compact surface of constant mean curvature which is bounded by a circle, but is not a piece of a sphere....

Show that any compact embedded CMC surface which is bounded by a convex planar curve, and lies on one side of the boundary plane, is topologically a d...

Show that the index of any singularity of a principal line fields on a surface is at most one....

Finding matching upper and lower bounds for k-sets and halving lines...