26 problems

Let $R$ be a Cohen--Macaulay ring and let $\mathfrak{p}$ be a prime ideal of finite projective dimension. Write … for the associated graded ring. Assume that $\mathrm G$ is a Cohen…

Let $Q=(q_n)_{n\geq 1}$ be a dynamically generated basic sequence, and let $\mathcal{UN}(Q)$ denote the set of uniformly $Q$-normal numbers. Uniform normality multiplication conjec…

Let $Q=(q_n)_{n\geq 1}$ be a nondeterministic dynamically generated basic sequence. Let $\mathcal{N}(Q)$, $\mathcal{DN}(Q)$, and $\mathcal{UN}(Q)$ denote the corresponding normalit…

Let $Q=(q_n)_{n\geq 1}$ be a deterministic dynamically generated basic sequence. Write $\mathcal{N}(Q)$ for the set of $Q$-normal numbers, $\mathcal{UN}(Q)$ for the uniformly $Q$-n…

Let $Q=(q_n)_{n\geq 1}$ be a deterministic dynamically generated basic sequence. Let the strongest hotspot theorem refer to the assertion that the hotspot hypotheses imply membersh…

For each $k\geq 1$, let $\langle\operatorname{sat}(k)\rangle$ be the associated word, and define the infinite word … An infinite word over a finite alphabet is normal if every word…

Simple-normality conjecture. The Fibonacci concatenation is simply normal in every base.

Absolute-normality conjecture. The Fibonacci concatenation is absolutely normal.

Normality conjecture for bases with four zeros. For every base $\beta$ with $\omega(\beta)=4$ and every nonnegative integer $y$, the Fibonacci concatenation is normal in base…

Uniform-distribution conjecture. For every base $\beta$ with $\omega(\beta)=4$, there exists a $K$ such that, for all $k\geq K$, the digits in the sequences…

Let an infinite sequence over a finite alphabet be formed either by concatenating the Fibonacci numbers in any base, by taking the Rudin–Shapiro sequence along the squares, or by t…

Attractor-generation conjecture. Given a hyperbolic IFS with probabilities, a sequence $\sigma$ is an element of $S$—that is, $\sigma$ generates the attractor of the IFS as describ…

A real $x$ is polynomially random if it satisfies the polynomial-time randomness notion used in the source, and it is rationally normal if it is normal in every rational base $r>1$…

Let $Q$ be a basic sequence. Hausdorff-dimension separation conjecture. If $Q$ is infinite in limit and $t$-divergent, and $\{1,\ldots,t\}=A\mathbin{\dot\cup}B$, then the set … has…

Suppose that $\mathbb{N}=A\mathbin{\dot\cup}B$ is a partition of the natural numbers. Let $Q$ be a basic sequence, and say that it is fully divergent when the relevant divergence c…

Let $t\geq2$ be an integer and suppose that $\{1,\ldots,t\}=A\mathbin{\dot\cup}B$. Let $Q$ be a basic sequence, and say that it is $t$-divergent when the relevant divergence condit…

Small-perturbation solvability conjecture. Whenever $\lVert\vec{\epsilon}\rVert$ is sufficiently small, this system has a solution

Let $Q=(q_n)_{n\geq 1}$ be a basic sequence that is infinite in limit and fully divergent of type I. A real number is $Q$-normal if it belongs to $\mathscr{N}(Q)$, and is AP $Q$-ab…

Let $\Xi$ index the strata $\Lambda_{(r_i),I}^{\xi}$ of the enhanced quiver variety $\Lambda_{(r_i),I}$, let $d_{(r_i),I}$ be its expected dimension, and let $\Lambda^\circ$ be the…

Let $\Lambda_{(r_i),I}$ be the enhanced quiver variety, let $\Lambda^\circ$ denote its specified open locus, and let $d_{(r_i),I}$ be its expected dimension. Dimension-bound conjec…

Let $\Lambda_{(r_i),I}$ be the enhanced quiver variety associated with the dimension data $(r_i)$ and index set $I$, and let $d_{(r_i),I}$ be its expected dimension. Normal complet…

The grid-graph conjecture. The cut polytope ${\rm Cut}^\square(G)$ is normal if $G$ is a grid graph.

The 4-connected plane triangulation conjecture. The cut polytope ${\rm Cut}^\square(G)$ is normal if $G$ is a 4-connected plane triangulation.

Sturmfels–Sullivant's conjecture. The cut polytope ${\rm Cut}^\square(G)$ is normal if and only if $G$ has no $K_5$ minor.

Zak's conjecture. 1. …