39 problems

Let $V$ be a subspace with coordinate projections $\pi_{E \setminus i}$, and let $Y_V \subset \prod_{i=1}^N \mathbb{P}^{n_i}$ be the associated Schubert variety. Suppose that the s…

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…

Generalized hotspot conjecture. Under these assumptions, $y\in\mathcal{UN}(Q)$. This would extend the strongest known hotspot criterion beyond the currently established hypotheses,…

Let $p>q\geq 1$ be coprime integers. An integer expansion is an element $u\in\mathcal{L}_{p/q}$, and let $\mathtt{wmax}_{p/q}(u)$ and $\mathtt{wmin}_{p/q}(u)$ denote the associated…

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…

Conjecture on removing the lower bound. The condition $|a_n|\geq\sqrt{8}$ for all $n\in\mathbb{N}$ can be removed from the conclusion that $\Lambda(\bfa)$ is transcendental; that i…

Hochster-configuration conjecture. The edge ideal $I(G)$ is normal if and only if $G$ has no Hochster configurations.

Let $X\subseteq\mathbb{P}^r$ be a nondegenerate projective variety of degree $d$ and codimension $e$. A variety is $n$-normal when the restriction map on degree-$n$ forms is surjec…

A Hochster configuration of a graph $G$ consists of two odd cycles $C_1,C_2$ satisfying: $C_1\cap N_G(C_2)=\emptyset$, where $N_G(C_2)$ is the neighbor set of $C_2$, and no chord o…

A real number is Borel normal in base 10 when each possible finite string of decimal digits occurs in its expansion with the limiting frequency expected for independent uniformly d…

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…

Randomness-extractor conjecture. There is a generalization of von Neumann's randomness extractor which computes normal reals from biased normal reals and $\lambda$-Martin-Löf rando…

Let $(a_n)_{n\geq 1}$ be the binary Ehrenfeucht–Mycielski sequence, and let $N_0(n)$ and $N_1(n)$ denote the numbers of $0$s and $1$s, respectively, among its first $n$ bits. Balan…

Existence conjecture. For every $t\in\mathbb{N}$, there is a solution $(c_1,c_2,\ldots,c_t)\in\mathscr{R}_t$ to the unperturbed system

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 the decimal digits of $\pi$, $e$, $\sqrt{2}$, and other analytically defined irrational numbers be viewed through their digit-counting statistics and normalized partial sums. L…