Mathematics Problem Archive

Is the complement of the essential ramification set in ${\mathbb{R}}^{d}$ a $K(\pi,1)$-space?...

A version of the previous problem, in which the complexity measure is not purely topological: namely, it is the lowest number of critical points of Mo...

Give an upper bound for the function $T\mapsto F$....

Is it true that any real Morsification of $f$ can be connected with one of complexity $\rho(f)$ by a generic path in the base of a versal deformation,...

What can be said about the number $\rho(f)$?...

If $M^n$ is a closed manifold with a strictly convex projective structure, is it cubulated?...

Suppose a hyperbolic $3$-manifold $M$ admits a CR structure, not necessarily a spherical one. Can the deformation theory of convex real projective str...

If $M$ is a closed hyperbolic $3$-manifold, is every projective structure on $M$ convex?...

Does there exist a nonzero polynomial on $U\otimes V\otimes W$ invariant under $SL(U)\times SL(V)\times SL(W)$ when $\dim U=\dim V=4$ and $\dim W=8$?...

Let $\beta$ be a nondegenerate bilinear form on a finite-dimensional real vector space $V$, and let $G=\operatorname{Isom}(\beta)\subset GL(V)$. Which...

Let $M$ be a $3$-manifold whose fundamental group acts on a simplicial tree without global fixed points. In the covering space associated to an edge s...

Does every closed hyperbolic $3$-manifold contain a closed $\pi_1$-injective surface with only double curves of intersection?...

Does every closed hyperbolic $3$-manifold contain a closed $\pi_1$-injective surface with the 1-line property: in the universal cover, every pair of p...

Are finite-volume hyperbolic $3$-manifolds virtually semi-fibered?...

Do closed hyperbolic $3$-manifold groups have finite-index subgroups that embed in a word-hyperbolic reflection group?...

Does every freely indecomposable cubulated hyperbolic group contain the fundamental group of a closed hyperbolic surface?...

If $M$ is a closed $3$-manifold, when does $\pi_1M$ act on the affine building for $SL(4,\mathbb{R})$ so that the quotient retracts to $M$?...

Do there exist fibered hyperbolic $3$-manifolds that are homology $S^2\times S^1$ and have arbitrarily large injectivity radius? Are there hyperbolic ...

Does every hyperbolic $3$-manifold have a finite-sheeted cover homeomorphic to a CAT(0) cube complex?...

Fix $\mu$ below the three-dimensional Margulis constant. For hyperbolic $3$-manifolds of volume less than $V$, drill all closed geodesics of length le...

If $M_1$ and $M_2$ are cusped hyperbolic $3$-manifolds, does there exist a cover $M'_1\to M_1$ and a nonzero-degree map $M'_1\to M_2$ taking cusps to ...

Give a rigorous explanation for why SnapPy works so well in practice....

Given $R>0$ and an integer $n\geq4$, is there an $\varepsilon>0$ and a finite set of hyperbolic $n$-simplices such that every closed cone $n$-manifold...

Let $M$ be a hyperbolic $3$-manifold and let $n$ be any positive integer. Does $M$ admit an ideal triangulation with $m\geq n$ positively oriented tet...

Which subsets of $S^2$ can occur as limit sets of convex-cocompact Kleinian groups?...

Characterize the limit sets of graph Kleinian groups and iterated graph-Kleinian groups. A graph Kleinian group is a convex-cocompact Kleinian group f...

Start with $n$ tetrahedra and glue their faces together at random. The vertex links need not be spheres but are essentially random triangulated surfac...

Rivin's experimental results appear extremely regular, possibly indicating additional structure. Investigate this phenomenon....

Let $f(x)\in\mathbb{Z}[x]$ be irreducible over $\mathbb{Q}$ with all roots real, let $f(\alpha)=0$, let $k=\mathbb{Q}(\alpha)$, and let $\mathcal{O}_k...

For $\mathcal{O}_k$ as in item 7.1, let $\Gamma_{\mathrm{unit}}$ have vertex set $\mathcal{O}_k$, joining two elements when their difference is a unit...

Can there be a degenerate Kleinian group that is not the fiber of a fibration and has algebraic trace field?...

Is there a singly degenerate Kleinian group for which all matrix entries of all group elements lie in one fixed number field?...

Do Haken hyperbolic $3$-manifolds have strongly irreducible Heegaard splittings?...

A band joining the components of a two-component link $L\subset S^3$ is called complicated if either its core cannot be isotoped to meet a splitting s...

Study the function Ψ: Mg→ [−log(µg),∞) as a (rational) Morse function. Classify its rational critical points. Deduce properties of the rational cohomo...

Letγ1,···,γ 2g be the standard basis of Z2g. Study the function L = ∑ Lγi:Yg→ [0,∞) as a Morse function. Find critical sets and deduce properties of t...

Work out the details of this construction of the completion Yg of Yg. Show that L:Yg→ [0,∞) extends to L:Yg→ [0,∞) and is a proper map. Ideally, inclu...

Find a (6g− 8)-obstructor complex L and a proper expanding map F:L× [0,∞)→T ≥ϵ g 3For concreteness we triangulate each sphere as the join of 0-spheres...

Show that there are many quasihomomorphisms f:MCG (Sg)→ R that satisfy (1) and (2) above plus (3) f is bounded on every G(q). I remark that the conseq...

(Ends spectrum). What are the possibile values of ends(Modg,H ) for finitely- generated subgroups H <Modg? It is well-known that the moduli space Mg h...

Compute CommModg (Γ) for various subgroups Γ< Modg. Paris-Rolfsen and Paris (see, e.g., [ Pa]) have proven that most subgroups of Mod g stabiliz- ing ...

(Volume spectrum). Determine for each 1≤ k≤ 3g− 3 the image of Volk: Xg(Γ)→ R. Determine the union of all such images as Γ ranges over all finitely pr...

Does there exist some Modg,g ≥ 2 that contains a subgroup Γ isomorphic to a cocompact (resp. noncocompact) lattice in SO(m, 1) with m≥ 5 (resp. m≥ 4)?...

(Holomorphic representatives). Find an algorithm or a group-theoretic invariant which determines or detects whether or not a given representation ρ: π...

(Normal subgroups). Let Γ be a finitely generated normal subgroup of Modg, whereg≥ 3. Must Γ be commensurable with Modg or with Ig? One way of constru...

Forg≥ 2, determine the irreducible factors of the graded pieces of the Malcev Lie algebra tg ofIg as Sp-modules. While Hain gives in [ Ha3] an explici...

Give a proof of Theorem 2.13 which does not depend on the classification of finite simple groups. 22 B. Farb To complete the picture, one would like t...

(Frequency of low symmetry). Let H denote the set of integers g≥ 2 such that N (g) = 8( g + 1). Find the s0 for which the series ∑ g∈Hg−s converges ab...

(Automorphism groups with special properties). LetP be a property of finite groups, for example being nilpotent, solvable, or a p-group. Prove a versi...

(Nonarithmetic extremal surfaces). Give answers to all of the above problems on automorphisms of Riemann surfaces for the collection of non-arithmetic...