17 problems

Let $S_n$ be the set of permutations of $\\{1,2,\ldots,n\\\\}$, and let two permutations be equivalent when one can be obtained from the other by replacing an occurrence of the pat…

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 $S$.

Let $X$ be a standard Borel space, and let $E$ be a Borel equivalence relation on $X$. Strong dualizability conjecture. Every such Borel equivalence relation is strongly dualizable…

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

The hyperfiniteness conjecture. Every countable amenable Borel equivalence relation is hyperfinite.

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…

Let $2^{\omega}$ be Cantor space. A function $f\colon 2^{\omega}\to 2^{\omega}$ is $(\equiv_T,\equiv_m)$-invariant if Turing-equivalent inputs have many-one-equivalent outputs; it…

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:

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

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

Given a composition $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_k)\models n$, define the permutation … Let $X_n=\{x_\alpha:\alpha\models n\}$. A crosshatch pair is a pair $x\leq w$ su…

Let $X$ be a smooth projective variety and let $C^i_{\sim}(X)$ be the group of codimension-$i$ cycles modulo an equivalence relation $\sim$. Write $C^i_{\sim_{\mathrm{hom}}}(X)$ fo…