27 problems

APN power-function classification conjecture. All APN functions of the form $F(x)=x^d$, defined over $\mathbb{F}_{2^n}$, are known up to CCZ-equivalence. This conjecture asserts co…

Let $adN$ denote the nonstandard natural numbers, let $qen$ be the equivalence relation used in the source, and let $Exp\dN$ be the exponential equivalence relation. A Borel equiva…

Correspondence conjecture. The Green's relations on a subsemigroup $U$ of $S$ have as counterparts prolonged images in $S$, namely the fine equivalence relations on $S$.

Two-class EA-equivalence conjecture. For every odd prime $p\geq 11$, these permutation polynomials fall into exactly two EA-equivalence classes. One class consists of the two speci…

An analytic equivalence relation is an equivalence relation on a Polish space whose graph is analytic. An analytic equivalence relation is universal if every analytic equivalence r…

Equivalence conjecture. If $(0,1,\lambda)\sim(0,1,\lambda')$, then either $\lambda$ and $\lambda'$ are both transcendental, or $\lambda$ and $\lambda'$ are conjugate algebraic numb…

Generic-classification dichotomy. Exactly one of the following holds:

Orthogonal Knuth equivalence conjecture. Two primed words satisfy

Cyclic c-equivalence conjecture. The set

Ma's alternating-pattern tightness conjecture. The bound $n\geq2c-1$ is tight exactly when $m$ is an alternating pattern.

Ma's rotational-equivalence cutoff conjecture. The cutoff satisfies $t=2c-1$ if and only if $m$ is alternating.

Kuszmaul and Zhou's conjecture. For a given pattern $m$, the number of nontrivial equivalence classes in $S_n$ for $n>|m|$ is independent of $n$.

Kotzig's conjecture. The maximum is

Let $(E_n)_{n\in\omega}$ be Borel equivalence relations on $2^{\omega}$ such that $E_n\subseteq E_{n+1}$ for every $n$, and suppose $\equiv_T=\bigcup_{n\in\omega}E_n$. A pointed pe…

Let $2^{\omega}$ be Cantor space. A function $f\colon 2^{\omega}\to 2^{\omega}$ is $(\equiv_T,\equiv_m)$-invariant, and a function is uniformly invariant when its many-one equivale…

Ding–Gao–Hjorth conjecture. If $E\leq_B E^X_G$ and $E$ is essentially countable, then $E$ is essentially hyperfinite.

Hjorth–Kechris–Louveau strictness conjecture. The following Borel reducibilities are strict:

Let $\cong_{f.p.}$ be the isomorphism equivalence relation on finite presentations of groups. It is uniformly finitely precomplete if every finite partial computable function into…

The conjecture. $PSP_{meager}({\mathbf{\Sigma^1_2}})$ is equivalent to the existence of Cohen generics over $L[z]$ for every real $z$.

Primitive-element conjecture. There exists an ergodic element $f \in [\mathcal{R}]$ such that the subequivalence relation generated by $f$ is primitive in ${\mathcal{R}}$.

Non-hyperfinite full-group conjecture. If $\mathcal R$ is not hyperfinite, then $[\mathcal R]$ does not have ample generics.

Completeness conjecture. $E$ might be complete under $\leq_c^4$, but not under $\leq_c^5$.

Thom's conjecture. There are no discrete free subgroups of rank $2$ in the full group of the hyperfinite ergodic equivalence relation.

Russell paradox conjecture. If $R\neq\varnothing$, then $R$ is the Russell Paradox.

Let $\mathcal{R}$ be a discrete, measure-preserving equivalence relation, let $c(\mathcal{R})$ denote its cost, and let $\beta^{(2)}_{1}(\mathcal{R})$ denote its first $l^{2}$-Bett…