10 problems

Let $\mathcal T_1$ be a classifiable theory and $\mathcal T_2$ a non-classifiable theory. A generalized Borel-reducibility main gap problem. asks whether there is a Borel reduction…

Let $E$ be a Borel equivalence relation, and let $E_1$ denote eventual agreement on countable sequences of reals. Write $E\leq_B F$ when $E$ is Borel reducible to $F$. An orbit equ…

Let $E$ be a Borel equivalence relation, and let $E_1$ denote eventual agreement on countable sequences of reals. For equivalence relations $E$ and $F$, write $E\leq_B F$ when $E$…

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

The analytic non-orbit-equivalence conjecture. The equivalence $E_1$ is not the least analytic equivalence relation which is not Borel reducible to an orbit equivalence.

For each $n\geq 1$, let $\cong_n$ be the isomorphism relation on the space of torsion-free abelian groups of rank $n$, and let $\cong_{n+1}^+$ be the corresponding relation with th…

For each $n\geq 1$, let $\mathcal{BD}^*_n$ be the class of Bratteli diagrams of exact rank $n$, and let $BD^*_n$ be the equivalence relation induced by equivalence of Bratteli diag…

Let $\bigsqcup_{n\geq 1} \cong_n^+$ denote the disjoint union of the isomorphism relations $\cong_n^+$ on the corresponding spaces of torsion-free abelian groups, and let…

Törnquist's conjecture. The isomorphism relation for separably acting factors is universal, with respect to Borel reducibility, for orbit equivalence relations induced by a continu…

Let $m,n\geq3$ and let $p,q$ be primes. Write $\oiso_{m,p}$ for isomorphism equivalence on the space of $p$-local torsion-free abelian groups of rank $m$, and $\oqiso_{n,q}$ for qu…