Mathematics Problem Archive

Prove that no infinite connected Ramanujan graph is Liouville; equivalently, every such graph admits a nonconstant bounded harmonic function....

Suppose $\mathbb{Z}$ acts on a graph $G$ by isometries, the quotient $H=G/\mathbb{Z}$ is Liouville, and simple random walk on $G$ visits every transla...

Let $G$ be the one-skeleton of a bounded-degree triangulation of an open disk. If $G$ is transient, prove that Bernoulli site percolation with paramet...

Tile the unit square by finitely or countably many squares of varying sizes, with at most three squares meeting at a corner, and color the squares ind...

Let $G$ be a bounded-degree triangulation of an open disk with polynomial volume growth. Prove that its Bernoulli site-percolation critical probabilit...

Let $G$ be the one-skeleton of a bounded-degree recurrent triangulation of an open disk. Prove that Bernoulli site percolation with parameter $1/2$ ha...

Let $G$ be the one-skeleton of a bounded-degree transient triangulation of an open disk. Prove that Bernoulli site percolation with parameter $1/2$ ha...

In the Poincaré disk, sample a Poisson process of intensity $\lambda$ with respect to hyperbolic area, form its Voronoi tessellation, and color cells ...

Let $(G_k)$ be a locally convergent sequence of bounded-degree graphs, each having separation profile of order at most the square root of the subgraph...

Construct the Poisson–Voronoi tessellation of the plane equipped with an $\ell_p$ metric and give the cells their adjacency graph metric. What is the ...

Delete each edge of the square lattice independently with probability $q<1/2$, condition the origin to lie in the infinite component, and let $K_q$ be...

Prove that there is a universal constant $C$ such that every finite connected vertex-transitive graph $G$ of degree $d$ satisfies $$R_{\mathrm{eff}}(u...

Among all finite connected vertex-transitive graphs rescaled by their diameters, which one minimizes Gromov–Hausdorff distance to the round sphere $S^...

Does there exist a family $(G_n)$ of finite $d$-regular graphs with $|G_n|\to\infty$ and a constant $h>0$ such that every induced metric ball in every...

Let the Euclidean plane or hyperbolic plane have a triangulation whose triangles have diameter at most $r$. Suppose that for every pair of radius-$r$ ...

Characterize the graphs that occur as tangency, or nerve, graphs of sphere packings with disjoint interiors in $\mathbb{R}^d$....

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 ...