Mathematics Problem Archive

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)...

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...

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...

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

Would it be more prudent to replace the conditionSc(g) > 0 byRicci> 0?...

Spin Problem. All of the above only applies to spin maps f ∶X→X, for which the required twisted Dirac operator defined, and, as on similar occasions we...

Conjecture: Stabilisation of Extremality.Let X0 be a compact area extremal Riemannin manifold. Then A. X0× Rm is area gap extremal for allm. 53 B. X0×...

Conjecture. The sphereSn minus Σo is area extremal in the "subcomplete" sense for all closed subsetsΣo ⊂Sn of topological dimensions k ≤1. 20 Lengths,...

Conjecture. All of the above is satisfied, modulo constants, for alln-manifolds, possibly incomplete and/or with boundaries, withSc(X) ≥σ > 0. Namely m...

Conjecture. Sharp Spherical Length Comparison Inequality. Spheres with finitely many punctures are length extremal. In fact – this is, probably equival...

Conjecture: ExtremalityofConcaveSphericalBalls. The balls B(R) ⊂Sn of radiiR≥π 2 are length extremal: no Riemannian metricg on such a ball which is gr...

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...