Mathematics Problem Archive

Let G ⊂ Isom(H4) be a Schottky group (or, more generally, a free convex-cocompact group). Can Hausdorff dimension of the limit set of G be arbitrarily ...

Let G be a finitely-generated discrete group of isometries of a Gromov-hyperbolic space X so that the limit set of G is connected. Is it true that the ...

Let $G$ be a group and let $\rho_1,\rho_2:G\to\operatorname{Isom}(\mathbb{H}^n)$ be discrete faithful representations; write $\ell_{\rho}(g)=\inf_x d(...

Find a “constructive” proof of the above theorem. More precisely, consider a finite presentation ⟨g1,.., gk|R1,.., Rm⟩ of G. Given [ ρ] ∈ D n(G) define ...

Find new restrictions on Kleinian groups. Recall that a group G is called coherent if every finitely-generated subgroup of G is finitely-presented....

Prove that every arithmetic lattice in Isom( Hn) ( n ≥ 4) is non-coherent. See [38] for some partial results in this direction. It is well-known that ...

Suppose that G ⊂ Isom(HHn) is a discrete subgroup satisfying Property T. Does it follow that G preserves a totally-geodesic subspace H in HHn and acts...

Suppose that ∆ is a developable triangle of groups, where all the cellgroups have Property T and so that all the links in the universal cover of T hav...

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

Generalize Vinberg’s finiteness theorem for reflection groups to complex-hyperbolic reflection groups, i.e., prove that there exists a number N such that...

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....

Generalize holomorphic chain patterns in ∂∞CHn in order to prove rigidity results for embeddings of lattices in P U(n, 1) into other higher rank Lie g...

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

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?...

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

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...

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...

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 ...

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?...

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

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...

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 $f,g$ 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 ...

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...

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....

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

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

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...

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...