21 problems

Let $G$ be a connected Lie group, let $ath.E(G,\mathcal{D}(G))$ denote the conjugacy relation on the space $\mathcal{D}(G)$ of discrete subgroups of $G$, and call an equivalence re…

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 $E$…

Amenability–hyperfiniteness conjecture. For countable Borel equivalence relations, Borel amenability is equivalent to hyperfiniteness.

Spectrum conjecture. For each countable ordinal $\alpha$, $=^{+\alpha}$ is in the spectrum of the meager ideal.

Countable invariant uniformization conjecture. Let $E$ be a Borel equivalence relation on a Polish space $X$. Then the following are equivalent: (a) $E$ is reducible to countable;…

A countable Borel equivalence relation is an equivalence relation on a Borel subset of $2^\omega$ whose equivalence classes are countable and whose relation is Borel. A Borel reduc…

Let $\mathbb E_1$ be the equivalence relation on sequences of reals defined by eventual agreement, and let a Borel equivalence relation be prime if it is Borel-reducible to no prod…

A countable Borel equivalence relation is an equivalence relation on a standard Borel space whose classes are countable and which is Borel as a subset of the product space. Marks's…

Nonuniversality conjecture. Recursive isomorphism on $2^\omega$ is not a universal countable Borel equivalence relation.

Treeable nonreducibility conjecture. The universal treeable countable Borel equivalence relation is not Borel reducible to Turing equivalence.

Optimistic uniform universality conjecture. If $E$ is a universal countable Borel equivalence relation, then $E$ is uniformly universal with respect to every way of generating $E$.

Uniform universality characterization. A countable Borel equivalence relation is universal if and only if it is uniformly universal with respect to every way it can be generated.

Essential hyperfiniteness conjecture. If

Let $\overline a$ and $\overline b$ be countable tuples from a monster model, with $\overline b$ containing $\overline a$. Let $E_L^{\overline a}$ and $E_L^{\overline b}$ be the co…

Let $T$ be a theory with monster model ${\EuFrak C}$, let $\overline{a}$ be a countable tuple from ${\EuFrak C}$, and let $E_L^{\overline a}$ and $F_L^{\overline a}$ denote respect…

Let $G$ be a group definable or $*$-definable over $\emptyset$ in a monster model, and let $E_G$ be the associated Borel equivalence relation coding the quotient $G^{00}/G^{000}$.…

Let $T$ be a first-order theory, and let $Gal_0(T)$ denote the kernel of the canonical map from the Lascar group $Gal_L(T)$ to the Kim--Pillay group $Gal_{KP}(T)$. Lascar-group non…

Let $X$ be an $E_{KP}$-class. The relations $E_L$ and $E_0$ are defined on spaces of types, where $E_L\!\upharpoonright\!X$ denotes the restriction of Lascar equivalence to $X$, an…

Let $\equiv_A$ denote arithmetic equivalence and $\equiv_T$ Turing equivalence on $2^\omega$. A homomorphism from $\equiv_A$ to $\equiv_T$ preserves equivalence, and uniformity mea…

Let $E$ be a countable Borel equivalence relation, and let $\equiv_A$ denote arithmetic equivalence equipped with the arithmetic cone measure. Arithmetic ergodicity conjecture. Exa…

Conjugacy-complexity dichotomy conjecture. If $cmathcal M$ is the countable model in an $caleph_0$-categorical theory, then $cmathcal C_{cmathcal M}$ is either smooth or Borel comp…