14 problems

Let $\mathbf{CH}$ be the category of compact Hausdorff spaces and continuous maps. A category is Stone-like when its abstract ultracoproduct construction satisfies the relevant Sto…

Let $d4aa$ be a free ultrafilter on $d4a9$ and let $d53d$ be a separable $d40fd40f_1$-factor. Let $d52c$ denote the hyperfinite $d40fd40f_1$-factor, and let $d52c^{d4aa}$ be its ul…

Equivalence conjecture. The restriction that the ultrafilter at a successor level be $\kappa_\alpha$-complete is unnecessary: the quotiented $\mathcal{F}$-hierarchies and ultrapowe…

Let $M$ be a structure whose first-order theory has the independence property witnessed by an affine quantifier-free formula. Let $(\Omega,\mu)$ be a diffuse probability space, and…

Let $\Gamma$ be the arithmetic group specified by the local-to-global conjecture above, fix a non-principal ultrafilter $\mathcal{N}$ on the natural numbers, and let $O^*$ and…

Let $N$ be a non-Gamma $\rm II_1$ factor with the Haagerup property. For an ultrafilter $\mathcal{U}$ on a set $I$, let $N^\mathcal{U}$ be the corresponding ultrapower, and let the…

Non-lifting conjecture. This phenomenon holds for any ultraproduct II$_1$ factor $\prod_\omega M_n$. The conjecture asks whether the failure of the lifting property established for…

Non-isomorphism conjecture. This assertion need not hold in ZFC for every such $M$ and every ultrafilter $\mathcal V$; moreover, ultrapowers of countable, elementarily equivalent s…

Let $\Phi$ be the homomorphism from automorphisms of the Weyl algebra to polynomial symplectomorphisms constructed by reduction modulo an infinite prime, restriction to centers, in…

Boolean-ultrapower saturation conjecture. There exist such $cappa$, $cB$, $U$, $cM$, and $cN$ for which

Let $(n_i)_{i\to\theta}$ and $(m_j)_{j\to\theta'}$ be families of natural numbers, and let $\theta$ and $\theta'$ be ultrafilters. Write $\tau$ for the corresponding ultralimit of…

Let $P$ be a separable subalgebra of an ultraproduct $\prod_{\mathcal U}N_i$ of $\mathrm{II}_1$ factors. The bicommutant amenability conjecture. If … then $P$ is amenable. This is…

Representable-ring ultraproduct conjecture. Every subdirectly irreducible representable $*$-regular ring can be embedded into a homomorphic image of a regular $*$-subring of an ult…

The depth inequality conjecture. In ZFC,