Mathematics Problem Archive

Study geometric properties of Markov spectrum....

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 least number of colors needed to color the Euclidean plane so that points at unit distance receive different colors....

Given a finite planar point set and a prescribed rectangle, approximate efficiently the minimum number of congruent copies of the rectangle needed to ...

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...

Given a polygonal room with perfectly reflecting sides and a point light source, characterize when every point of the room is illuminated. In particul...

Can the $3\times6$ rectangle with a $2\times2$ corner removed tile a rectangle using an odd number of congruent copies?...

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?...

Can sums of square roots of integers be compared in polynomial time on a Turing machine? Equivalently, obtain effective polynomial bit bounds for a no...

Find the largest function $t(n)$ such that every convex $n$-gon has a triangulation containing at least $t(n)$ distinct triangle areas; also determine...

Can the number of simple polygonalizations of a set of $n$ points in the plane be computed in polynomial time? A simple polygonalization is a simple p...

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

Consider minimal immersions $S^n\to S^N(1)$ with total area at most a fixed constant $A$, identifying immersions which differ by a motion of the ambie...

Let $M$ be a compact surface, $I=(-1,1)$, and $f:M\times I\to\mathbb{R}^3$ a differentiable map such that each $f_t$ is an immersion and its induced m...

Let $B$ be a set of $n+2$ hyperplanes in general position in complex projective space $\mathbb{P}^n(\mathbb{C})$. Prove that there is no nondegenerate...

Let $\Delta$ be a ball in $\mathbb{C}^n$, $n\geq2$, and let $F$ be a complete Hermitian manifold. Does every holomorphic map from $\Delta\setminus\{0\...

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...

Are braid groups CAT(0)?...

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...

There exist values of $\lambda_1$ and $\lambda_2$ such that the spectrum $\sigma(H)$ of $H$ is a Cantorval; that is, the spectrum is the closure of it...

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 $\lambda$....

Determine the position of sharp rings in the diffraction measure of a Pinwheel Tiling and their intensity....

Does the diffraction measure of the Pinwheel Tiling contain an absolutely continuous component?...

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 $p(n)$ count radius-$n$ patches in an aperiodic repetitive tiling of dimension $d$, and let $\Omega$ be its tiling space. If $p(n)=O(n^d)$, must t...

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...

Classify tilings having a geometric property such as bounded-displacement equivalence (BD), bi-Lipschitz equivalence (BL), or linear repetitivity (LR)...

Develop and study new geometric properties, analogous but not identical to BD, BL, etc., that are invariant under MLD, topological conjugacy, or homeo...

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...

Let $B=B\Gamma$ be the classifying space of a discrete countable group, and let $f:X\to B$ be a continuous map from a Riemannian manifold. Does there ...

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

For instance, ifX =SO(n) withn≥5, then no known method can rule out metricsg ≥g on X with Sc(g) > Sc(g)....