Mathematics Problem Archive

there is no apparent non-trivial bound on the width ofX = Σn−1× [−1, 1]even we assume that thesectional curvatureof X is = 1....

Also it is unclear if there are (non-trivial) inequalities of this kind for other exotic spheres....

But it is also not impossible that all manifolds admit immersions into the unit ball in the Hilbert spaceR∞with principal curvatures bounded by a univ...

Problem. Identify combinatorial typesPextr of convex polyhedra where all representativeP ∈P are extremal and also describe extremal P of non-extremal ...

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. Shrinking of Singularities. Let X be a compact orientable Riemanninn-manifold, f0 ∶X →Tn be a continuous map of non-zero degree, hi, i=0,1...

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. Parametric Hypersphericity. Let X be a complete oriented Riemanniann-manifold and letΨ(X) ⊂Lipλ(X →Sn(1))) be the space of 1-Lipschitz loc...

If m = n−1 then, conjecturally, this is the only manifold with this property: the inequalities macr.dim(Ψ(X)) ≥1 and Sc(X) ≥(n−1)(n−2) should imply th...

Conjecture. Stability of Periodic Slabs. The only Zn−3-invariantmeanconvexdomainsin Rn withdisconnectedboundaries are slabs between parallel hyperplan...

Conjecture. C0-closeness of the spaces ofC0-metrics withVolumicallyPositiveScalarCurvatures. IfaRiemannian C0-metricg onan n-dimensionalmanifold X can...

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

Conjecture. Non-Riemannian Guth-Geroch. Let X be an n-dimensional Q-essential pseudomanifold (e.g. manifold) with an arbitrary metric. Then the univer...

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

Conjecture. Geroch for Alexandrov Spaces.If anndimensionalAlexandrovspace X withsect.curv ≥−1andScvoln(X) ≥ 0 admits a continuous mapΦ with non-zero d...

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...

Write (if exist) Cayley algebra systems defining the strata for the following graph: Currently this example is a strong candidate for a counterexample ...

Develop theory of geometric conditions for strata in multidimensional case....

Does there exist a closed C2 surface in Euclidean space R3 which is flexible?...

Are all smooth tight surfaces in R3 rigid?...

Given a metric of positive curvature on the disk what is the condition on a space curve to form the boundary of an isometric embedding of the disk?...

Are there some nonconvex surfaces which remain rigid after finitely many points of them have been deleted. For instance, are punctured analytic tight s...

(The global isometric embedding problem, Yau [189] 1993; Gromov [82]). Can everyC∞ 2-dimensional Riemannian manifold be isometrically embedded in R4?...

For which setsA⊂ Sn is there an immersionf: M→ Rn+1 such that Gf(M)⊂A?...

LetM,M′⊂ R3 be smooth orientable closed surfaces. Suppose there exists a diffeomorphism f: M→ M′ which preserved the Gauss curvature and the Gauss map....

Does there exist a convex polyhedron with a pseudo edge graph which is not unfoldable....

Let $\Gamma$ be a closed curve of fixed length $L$ in $\mathbb{R}^3$. Determine the maximum possible volume of the convex hull of $\Gamma$....

Is every compact connected minimal surface bounded by a pair of convex planar curves topologically an annulus?...

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...

Are there any complete surfaces of negative curvature in Euclidean 3-space whose principal curvatures are bounded away from zero?...

Are there any complete negatively curved surfaces embedded in the unit ball?...

Does there exist any complete negatively curved surfaces with negative Euler characteristic contained in between a pair of parallel planes in R3....

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...

Determine the automorphism group of the partial order of Turing degrees....

Is the complement of the finite spectrum of every first-order sentence also a finite spectrum? Equivalently, is $\mathrm{NE}=\mathrm{coNE}$?...

Does there exist a reasonable logic strictly stronger than first-order logic that has both compactness and Craig's interpolation property?...

Prove a superpolynomial lower bound on the size of Frege proofs; in particular, do some tautologies require exponentially large Frege proofs?...

Determine the reverse-mathematical strength of Kříž's labeled-tree generalization of Kruskal's theorem....

The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for $\aleph_1$-saturated models of a countable theory....

Shelah's categoricity conjecture for $L_{\omega_1,\omega}$: If a sentence is categorical above the Hanf number then it is categorical in all cardinals...

Shelah's eventual categoricity conjecture: For every cardinal $\lambda$ there exists a cardinal $\mu(\lambda)$ such that if an AEC K with LS(K)${} \le...

Does every simple first-order theory have stable forking?...

The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under ...

The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?...

Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality $\aleph_{\omega_1}$ d...