Mathematics Problem Archive

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. C2-Smoothing of Continuous Metrics with Volumically Positive Scalar Curvatures.All continuous Riemannian metricsg on a smoothn-dimensional...

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 $\operatorname{Sc}(X)\geq-\sigma^2$ ...

Can one hear the presence of an orbifold singularity, i.e. whether or not there exists a pair of isospectral orbifolds, one of which has singular poin...

Does $S^2\times T^2$ admit a Riemannian metric with positive biorthogonal curvature?...

For ${2n+1}>3$ and $0<k<{2n+1}$, are projective subspaces obtained by projection of the $k$-dimensional equatorial spheres minimal submanifolds of the...

For solvable Lie groups $G$, show that solvsolitons are the only local maxima of the Ricci pinching functional $g\mapsto F(g)=\frac{Scal(g)^2}{|Ric(g)...

; Do there exist embedded minimal surfaces with finite genus and Morse index $4$?; More focussed: in the $1$-parameter deformation of Costa's surface,...

- Construct congruence lattices which are norm$_1$ and norm$_1*$- isospectral in all dimensions (see ). - Are there families of $p$-isospectral lens s...

Prove that geodesic spheres provide the least-perimeter way to enclose prescribed volume in $CP^2$....

1. If the normal holonomy group of an irreducible and full homogeneous submanifold $M^n$ of the sphere with $n \geq 2$ does not act transitively, then...

; Do CMC spheres about a point $x$ in such a space form a foliation of $X-\{x\}$?; Could this be a way of proving embeddedness of CMC spheres in gener...

Does the normalized Ricci flow converge in Gromov-Hausdorff sense to a generalized K\"ahler-Einstein space?...

Does Frankel's theorem hold for symmetric Finsler metrics?...

Describe the combinatorics of B2(K6); B3(K4) and B3(K5)....

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...

Let Γ be a smooth closed curve immersed in R3. Suppose that Γ has a continuous binormal vector field B which is one-to-one. Does it follow then that th...

Given a C∞ metric in a neighborhood of a point in a 2-dimensional Riemannian manifold, does there exist an isometric embedding of some neighborhood of...

What is the shortest curve in R3 with a given width or inradius?...

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....

Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets...

Square packing in a square: what is the asymptotic growth rate of wasted space?...

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

Tripod packing: how many tripods can have their apexes packed into a given cube?...

Dissection into orthoschemes – is it possible for simplices of every dimension?...

The values of the Hermite constants for dimensions other than 1–8 and 24...

The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solutio...

Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one...

Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?...

Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram?...

What constant $1.1 < a \leq 10.76$ governs the lower bound of a closed knot $K$'s minimum ropelength $L(K) \geq a\operatorname{Cr}(K)^{3/4}$?...

Is the upper bound of a closed knot's minimum ropelength linear to its crossing number?...

The Thomson problem – what is the minimum energy configuration of $n$ mutually-repelling particles on a unit sphere?...

Let $N$ be the closed $3$-manifold obtained by gluing two copies of the figure-eight knot complement along their torus boundaries by a homeomorphism. ...

Let $N$ be a closed $3$-manifold whose JSJ decomposition contains only hyperbolic pieces. Does $N$ admit a convex projective structure?...

Geometric transitions are continuous paths of geometries that abruptly change type in the limit. Understand all transitions between the eight Thurston...

The renormalized volume of quasi-Fuchsian groups gives a function $\rho:\mathcal{T}(S)\times\mathcal{T}(S)\to\mathbb{R}$. Is $\rho$ a metric on the Te...