Mathematics Problem Archive

Is there are meaningful structure theory for lacunary hyperbolic groups? Can one define a useful boundary for such groups? Is it true that either Out(G...

Every relatively hyperbolic group has cut points in all of its asymptotic cones. To what extent does the converse hold? Characterize the finitely gene...

For the fundamental group G of a closed hyperbolic n-manifold consider a short exact sequence 1 →Zp → Γ → G → 1. Is the group Γ residually finite? In o...

Let G be a Gromov-hyperbolic Coxeter group. Does G admit a discrete embedding in Isom(Hn) for large n?...

Let G ⊂ Isom(Hn) be a discrete torsion-free finitelygenerated subgroup without abelian subgroups of rank ≥ 2. Is it true that (a) cdZ(G) ≤ Hdim(Λc(G)) ...

Generalize Bestvina-Feighn combination theorem from graphs of groups to complexes of groups....

There is a theory of quasi-convex groups acting on Gromov hyperbolic spaces, generalizing the theory of convex-compact groups of isometries of the rea...

Extend this relation of Anosov structure and dynamics on the limit set to representations of other hyperbolic groups....

Obtain new rigidity results for embeddings of realhyperbolic lattices into higher-rank semisimple Lie groups in terms of the boundary maps....

Consider Teichm¨ uller spaceT (S) with Teichm¨ uller metric. Does it have quadratic isoperimetric inequality?...

Compute cogrowth for notable subgroups $H\subset G$, where cogrowth is the growth of the Schreier graph $\Gamma_{G/H}$. In particular: prove that the ...

If knots $K$ and $K'$ have the same Alexander polynomial, are the corresponding Fintushel--Stern four-manifolds $X_K$ and $X_{K'}$ diffeomorphic?...

Extend Liu's classification of compact symplectic four-manifolds with positive numerical invariant $\kappa$ to the borderline case $\kappa=0$; determi...

Is a symplectic structure $\omega$ on a four-manifold unique up to diffeomorphism when the elementary topological invariants $[\omega]$ and $c_1(M)$ a...

Prove an appropriate complex analogue of the Jörgens--Calabi--Pogorelov theorem: classify global solutions on $\mathbb{C}^n$ of the complex Monge--Amp...

Which compact seven-manifolds admit a Riemannian metric with holonomy $G_2$?...

For a compact seven-manifold $M$ admitting holonomy-$G_2$ metrics, describe their moduli space modulo diffeomorphisms isotopic to the identity. If $\p...

Develop compactness and singularity theories for special Lagrangian, associative, and co-associative calibrated submanifolds that are strong enough to...

Let $\mu(n)$ be the least $m$ such that a convex plane curve traversed $m$ times has a regular small perturbation in $\mathbb{R}^n$. Determine $\mu(n)...

For incompressible Navier-Stokes on $\mathbb{T}^d$, $d=2,3$, is the system approximately controllable and/or controllable in finite-dimensional projec...

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...

○What arepossible topologiesof manifolds whichadmit Riemannin metrics with scalar curvaturesSc > 0?...

○What are topologies ofspaces of metricsg with Sc(g)>0?...

○What are geometries ofindividual manifoldswith Sc > σ?...

○What are effect of lower boundsSc ≥σ on the topology and geometry of maps between manifolds?...

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...

Conjecture that the surface-tangent-bundle condition in Llarull's scalar-curvature rigidity theorem is redundant. Specifically, let $X$ be a closed or...

Find a useful local geometric definition of a scalar-curvature lower bound $\operatorname{Sc}\geq\sigma$ that supports global theorems and extends to ...

Problem Extend the concept ofSc > 0 to singular Fano Varieties. For example, work out a definition ofSc(X) along the lines suggested in Question 1 of t...

Conjecture: Q-Non-Essentiality of Manifolds with Sc > 0. No rational homology class14 in the classifying spaceBΓ of a discrete groupΓ can be realised ...

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

Conjecture. Singularities are Unstable. Brian White told me about 30 years ago that he believed that Volume minimising hypersurfaces in generic Rieman...

Conjecture 6. ISC: Singularities are Irrelevant. Schoen and Yau announced 35 years ago [110], [114] that their descent metod extends to singular minim...

Let $X_{\mathrm{fl}}=\mathbb{R}^n/\Gamma$ be a complete flat manifold whose group $\Gamma$ acts by parallel translations. If a complete Riemannian man...

Probably, one can fully determine assumptions onπ1(X) depending on Xfl needed for this conclusion for all flat manifoldsXfl....

Also one can possibly relax theisometry at infinitycondition by some "asymptotic flatness" and negativity of a suitable "energy at infinity"....

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

Besides, similar results are expected for allsingular Alexandrov spaces with lower curvature boundsbut the Almgren’s regularity theory has not been de...

Conjecture: Waist-Width Inequality. All complete Riemannian n-manifolds X satisfy widthn−1(X) ≤constn⋅waistn−k+1(X). Contractibility Radius. This "rad...