Mathematics Problem Archive

Prove that every sphere packing in $\mathbb{R}^3$ whose tangency graph is $\mathbb{Z}^3$ has at most one accumulation point in the one-point compactif...

On the three-branch hierarchical graph $G_n$ described in Section 9.3, launch external-DLA particles from the sink until a particle settles at the sin...

In the random hierarchical graph obtained by repeatedly replacing a uniformly chosen edge by the fixed three-edge pattern of Section 9.4, let $D_n$ be...

Start from one edge and at each stage replace every edge independently by two edges in series with probability $p$ or two edges in parallel with proba...

Does there exist a mixing random tiling of the Euclidean plane whose law is invariant under rotations and translations, is stationary under a local cl...

For which dimensions $d$ can $\mathbb{R}^d$ be partitioned into pairwise disjoint curves, each of which has the law or geometric regularity of a Brown...

Run two walks with a common clock on $\mathbb{Z}^d$, each choosing uniformly among neighbors not previously visited by the other walk. Prove that in $...

Generate a quadrangulation from $2n$ bits using the Schaeffer bijection and independently resample each bit with probability $\varepsilon$. Determine ...

Let $G$ be a bounded-degree recurrent planar triangulation with a fixed root. Are there arbitrarily large $r$ and finite domains containing the radius...

Let $(G_n)$ be finite transitive graphs with $|G_n|\to\infty$ and uniformly bounded effective resistances between all vertex pairs. Prove that the lar...

For every $C<\infty$, prove that there is $c=c(C)<1$ such that, for every simple $n$-vertex graph $G$, the probability that simple random walk covers ...

Let $G$ be a bounded-degree triangulation of a disk. Prove that each of the following conditions implies $p_c(G)\le1/2$: $\operatorname{Dim}(G)\ge2$; ...

Let $G$ be a planar graph and consider Bernoulli percolation at $p=1/2$. If an infinite open cluster exists almost surely, prove that there are almost...

For a quasi-transitive graph $G$, characterize when Bernoulli percolation has a unique infinite cluster at $p=p_u(G)$. In particular, give necessary a...

On $\mathbb{Z}^d$, start with the origin black and every other vertex white. Repeatedly choose uniformly an edge having one black and one white endpoi...

Let $(X_n)_{n\in\mathbb{Z}}$ be a centered strictly stationary sequence with $\mathbb{E}[X_0^2]<\infty$. For $k\ge1$, define $$\phi_X(k)=\sup_m\sup\bi...

Is there an effective algorithmic procedure to produce and recognize a hyperbolic knot of depth $n$ for any given $n$? What about $\ge n$?...

Given a collection $\mathscr{C}$ of topological or geometric types of surface, what $3$–manifolds admit a taut foliation $\mathscr{F}$ whose leaves ar...

Let $X$ be a vector field on a $3$–manifold. When is there a foliation $\mathscr{F}$ of $M$ transverse to $X$?...

Generalize the Teichmüller polynomial from the fibered faces of the Thurston norm ball to the other faces (of some possibly generalized polyhedron, pe...

Given a collection of taut foliations $\mathscr{F}_i$ of $M$, what are the obstructions to finding a metric on $M$ for which the $\mathscr{F}_i$ (afte...

How many Reeb components must a foliation of an open $3$–manifold contain?...

What generalizations of the notion of taut foliation make sense on an open $3$–manifold?...

Characterize those essential laminations which contain genuine sublaminations....

Suppose $\Lambda$ is a full genuine lamination; i.e. it has some complementary region which is an ideal polygon bundle over a circle. Suppose $M$ is h...

Suppose $\Lambda$ is a genuine lamination. When can $\Lambda$ be ``filled in'' to a very full lamination $\Lambda'$? Does it help for $M$ to be hyperb...

Are loosesse laminations good for anything? Are leaves of the universal cover of a loosesse lamination properly embedded? If $M$ contains a loosesse l...

Give an example of a lamination in an atoroidal manifold –- perhaps loosesse –- which can never be realized by minimal surfaces for any metric, but wh...

Characterize branched surfaces embedded in $3$–manifolds which can be non–trivially split to a homeomorphic copy of themselves....

Develop a theory of hierarchies for branched surfaces....

When does a Haken sum operation make sense for a pair of laminations in normal form with respect to a fixed triangulation?...

Let $M$ be a $3$–manifold, and $\Lambda$ an essential lamination. Let $C$ be a cycle representing the fundamental class of $M$. Is there a cycle $C'$ ...

Suppose $\mathscr{B}$ is a branched surface in $M$ which is dual to a taut local orientation. Is there a finite cover of $M$ in which the pullback of ...

Give a useful definition of thin position for an embedded graph $\Gamma \subset M$ with respect to a taut foliation $\mathscr{F}$. If $\Gamma$ is the ...

Suppose $\mathscr{F}$ is an $\mathbb{R}$–covered foliation of an atoroidal $3$–manifold $M$. Is the holonomy representation $\rho_H$ of $\pi_1(M)$ on ...

For a fixed manifold $M$, describe the structure of the set of all essential laminations with a transverse $\widetilde{SL(2,\mathbb{R})}$ structure....

Suppose $M$ admits a minimal taut foliation. What is the best analytic (transverse) quality of a taut foliation it admits? Can we find a minimal folia...

Is there a universal constant $c$ such that a hyperbolic $3$–manifold $M$ whose fundamental group $\pi_1(M)$ can be ordered out to radius $c$ can be l...

Let $\mathsf{T}$ be some class of abstract computers; e.g. finite state automata, Turing machines, Turing machines relative to some oracle $O$, etc. A...

Let $\Lambda^\pm$ be a pair of laminations of $S^1$ which are transverse to each other and have finite area complementary domains. Suppose $\Gamma$ is...

What is the best analytic quality for the action of $\pi_1(M)$ on a universal circle $S^1_\mathrm{univ}$?...

Is there a universal transverse surgery description of tautly foliated manifolds, in the sense that there is a fixed $M$ such that for every tautly fo...

Give a collection of fundamental operations on foliations and an explicit family of base foliations such that every tautly foliated manifold $M,\maths...

Suppose $K$ is a non–torus alternating knot. Then essential laminations can be constructed which realize every (nontrivial) boundary slope. Can essent...

It is known that if a $3$–manifold $M$ contains an essential surface of genus $g$, the distance of any Heegaard splitting of $M$ has distance at most ...

Suppose $\mathscr{F}$ is a taut foliation of a hyperbolic $3$–manifold $M$ with two–sided branching. Must there be a leaf $\lambda$ of $\widetilde{\ma...

Suppose $\mathscr{F}$ is a finite depth foliation of a hyperbolic $3$–manifold. What is the relationship (if any) between the Hausdorff dimension of t...

What do short geodesics look like with respect to taut foliations? Is there a universal $\epsilon$ such that for every hyperbolic manifold $M$, every ...

Is there a uniform bound on the Godbillon–Vey invariants of the taut foliations of a hyperbolic manifold in terms of its volume?...

Suppose $\mathscr{F}$ is a taut foliation of a hyperbolic $3$–manifold $M$. Let $$\pi:\widetilde{M} \to L$$ be the projection to the leaf space of $\w...