Mathematics Problem Archive

Let $K(X,Y,Z,W)=h(R(X,Y)Z,W)$ be the curvature $(0,4)$-tensor of a statistical manifold. Investigate the structure of all $(0,4)$-tensors satisfying $...

On a statistical manifold, let $D$ be the rank-$(n-1)$ distribution orthogonal to the velocity of the $\nabla$-geodesic from a fixed point. If every s...

Estimate the minimum number of inflection points of a closed curve on a Riemann surface with a projectively flat connection....

For a closed curve on a Riemann surface with projectively flat connection, estimate the number of affine vertices and, assuming isolated inflection po...

Prove existence of a time-global solution to the affine-plane curve evolution equation $\partial x/\partial u=(1+kp)x''$, where $u$ is time, $x''$ is ...

If $(M,g)$ is a complete Hessian manifold, is its tangent bundle with the natural complex structure a Stein manifold? In particular, if a convex domai...

Let $M$ be a compact Hessian manifold and deform its affine structure. Does every sufficiently small deformation admit a Hessian metric?...

Construct compact Hessian manifolds other than direct products of hyperbolic and flat compact Hessian manifolds; in particular, construct one with no ...

What spaces can arise as boundaries of hyperbolic groups? As a sub-problem: For which k do k-dimensional stable Menger spaces appear as boundaries?...

What 2-dimensional spaces arise as boundaries of hyperbolic groups?...

Find further universality phenomena: classes of groups or spaces of different nature whose ideal boundaries are nevertheless all homeomorphic, beyond ...

Extend the known necessary conditions for a compact metrizable space $X$ to be the boundary of a proper cocompact CAT(0) space, or give a complete cla...

Does every CAT(0) group have finite asymptotic dimension?...

Develop a theory for analysis on the ideal boundaries of relatively hyperbolic groups, as it is done for hyperbolic groups....

In what generality does quasiconformal imply quasisymmetric?...

Are all quasi-isometric embeddings between higher-rank symmetric spaces either isometries or algebraic in this way?...

[Cannon–Thurston] Is this action conjugate to a conformal action?...

Let H ⊂ G be a hyperbolic subgroup of a hyperbolic group (we do not assume that H is quasiconvex). Is it true that there exists an equivariant continu...

What is the Poisson boundary of the free group with an arbitrary measure?...

Characterize the hitting measure ν on PMF obtained from the random walk by mapping classes on Teichm¨ uller space. Is it absolutely continuous with re...

What is the Poisson boundary of Outer space?...

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)...

○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?...

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 ...

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...