Mathematics Problem Archive

If X is a compact polyhedron and G is a discrete group of simple homotopy equivalences X → X, is there a compact space X ′, homotopy equivalent to X, ...

Consider finite cell complexes X. Is there an algorithm to determine if X is contractible?...

For a word-hyperbolic G not splitting over any virtually cyclic group, can an infinite-index subgroup and a finite-index subgroup be isomorphic?...

Is there a similar statement to this inflexibility result this with no group specified—that is, for subsets Λ ⊂ S2 of the boundary sphere of H3?...

Given p ∈ H3, estimate the biLipschitz constant of QΛ near p in terms of the distance d from p to the exterior of the convex hull of Λ. More concretel...

Are braid groups CAT(0)?...

Extend Rips’ theory to higher-dimensional buildings, e.g. products ofR-trees. Rank rigidity. Let X be a CAT (0) metric space. The space X is said to b...

Suppose that Y is a compact finitedimensional locally CAT (0) metric space of rank n ≥ 2. Then either the universal cover of Y splits (nontrivially) as...

Under the above assumptions, is it true that Y has coarsely trivial πm for m ≥ 2?...

Does the Coarse Whitehead Conjecture hold if G is hyperbolic?...

Is there a useful list of surgery procedures which generates all smooth four-manifolds of a given homotopy type?...

Find a structure or conjectural decomposition for smooth four-manifolds that could play the guiding role that Thurston's Geometrization Conjecture pla...

Let $f,g$ be smooth vector fields on an $n$-dimensional manifold $M$, and consider $\dot q=f(q)+ug(q)$, $|u|\leq1$, with fixed endpoint. For a generic...

Let $\gamma_i:[0,1]\to M$, $i=0,1$, be smooth admissible paths of a sub-Riemannian structure with $\gamma_0(0)=\gamma_1(0)=q_0$ and $\dot\gamma_0(0)\w...

For the endpoint map from the $H^1$ Hilbert manifold of admissible paths starting at $q_0$ to $M$, can the singular curves starting at $q_0$ fill all ...

Find a $C^1$-classification of the germs of sub-Riemannian spheres at points of optimal singular curves for generic metrics. In particular, obtain suc...

A distribution is singular transitive if any two points can be connected by a concatenation of singular curves. Does singular transitivity imply that ...

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...

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 $p(n)$ 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...

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...

But deeper structures (if they exist at all) that lie at the roots of Dirac operators and of minimal hypersurfaces are yet to be revealed....

Identify the most general classes of geometric objects having properties analogous to those of $C^2$ Riemannian manifolds with $\operatorname{Sc}\geq\...

Question. What could be a, possibly non-geometric, extension of the concept ofSc ≥0, where one would be able perform symmetrization and reduce the cas...

Question. How common are Ricci flat metrics on compact simply connected manifolds X which admit metrics with positive scalar curvatures?...

Conjecture. Connected sums of sufficiently many copies of compact manifolds that are not homotopy spheres carry no Ricci-flat metrics....

Problem. Remove the spin an the uniform positivity conditions, relax completeness and determine the sharp value ofconstn (depending on theK-theory cla...

It seems not impossible, at least for compactX, that, in fact, K-area(X×R) =K-area(X×S1)....

Question. Is the residual finiteness of the fundamental group essential?...

(i) It is unclear if the last step in the above argument is truly needed: conceivably, maps Φ ∶Sn−1→U(N) with Lip(Φ) < 1 2 are contractible to constan...

(iii) TheSn- andS2-product inequalities seems to hold fornon-trivial sphere fibrations, but I have not checked this carefully....

On the other hand, if theδ-neighbourhoods Uδ(S) ⊂X of allT(X)- non-spin surfacesS in a Riemannian manifoldX are "large" then the spin area of X must b...

For instance, letX be homomorphic to CP 2 and letvol(Uδ(S)) ≥δ2 for allT(X)-non-spin surfacesS ⊂X and 0<δ ≤1. Is then spin-area(X) ≥1/1 000 000?...

On the other hand, there probably exist compact simply connected n-dimensional manifolds for alln ≥4 with arbitrarily prescribed (finite) values of the...

the sharp values ofwidthn−m for these solids remains problematic for m≥2, (unless I missed some paper)....

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 ...

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

Are there compact manifoldsX which support metricsg withSc(g) > 0 but admit no area extremal or length extremal metricsg?...

Can one "effectively" evaluate the minimal constantλ=λ(X,g)„ such that a given Riemannin manifoldX = (X,g), e.g wheresect.curv(g) > 0, would support an...

But it is unclear if this remain true with "area" in place of "length"....

Question. When does such anX0 is area extremal in the category of complete manifolds?...

Extension Problem.LetX be a Riemanniann-manifold withSc(X) ≥σ > 0 and letσ−≤σ,r andr+ ≥r be positive numbers. Whendoesthereexistan n-dimensionalmanifo...

Conjecture. Completion by Extension.If σ > σ−and r ≥constn(σ −σ−)−1 2 for some (large) constant constn, then the extension problem is solvable withr+ ...

Conjecture 18. Interior Hemi-Spherical Area Inequality. The r-interiors of all compact Riemanninn-manifolds X with boundaries and withSc(X) ≥Sc(Sn) =n...

What are possible values of the co-s+areas of the complements ofr-balls in compact spin manifoldsX withSc(X) ≥n(n−1)?...

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

what is the (asymptotically forn→∞and/or fork→∞) sharp inequality for immersions of theseΣn−1 to spheres...

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