42 problems

Let $X$ be a Tychonoff topological space, let $F_{\mathrm{SIN}}(X)$ be the free balanced topological group on $X$, and, for a Hilbert space ${\mathcal H}$, let $U({\mathcal H})_u$…

Unitary representability conjecture. For every Tikhonov space $X$, there exists a metric space $M$ of $L^1$-type such that $F(X)$ is isomorphic to a subgroup of the group of isomet…

Analytic continuation conjecture. The function $\zeta_{\mathrm{ts}}(s;G,H)$ admits an analytic continuation to the complex plane, with poles reflecting the arithmetic structure of…

Modular distribution of invariants. The invariants of topo-symmetric extensions are distributed uniformly modulo $\gcd(|G|,|H|)$ in the sense above. Numerical experiments for small…

Let $P$ be a set in the relevant Euclidean space, and let $A$ be a Baire set. An affine image of $P$ is a set obtained from $P$ by an affine transformation. Topological Erdős simil…

A topological group is CDH (countable dense homogeneous) if any two countable dense subsets are carried to one another by a homeomorphism of the group. A space is crowded if it has…

Recall that $\aleph_{1}$ denotes the first uncountable cardinal, and let $D_{\aleph_{1}}$ be the discrete space of cardinality $\aleph_{1}$. The space…

Let $X$ be a connected manifold, let $\cl G$ be a closed subgroup of $\operatorname{Homeo}(X)$, and let $f\in\cl G$ be almost periodic, meaning that the subgroup…

Van Engelen's conjecture. The same equivalence holds for every zero-dimensional Borel space that is not locally compact.

Let $G$ be a locally compact group, let $K$ be a compact large subgroup of $G$, and let $S$ be a $K$-space. The neighborhood equivariant retract conjecture. $S$ is a neighborhood…

Let $G$ be a compact topological group. A subgroup $H\leq G$ is dense if its closure in $G$ is $G$, and $d(H)$ denotes the density character of $H$, while $w(G)$ denotes the weight…

Let $\mathcal F:\mathbb R\curvearrowright(X,\mu)$ be a measure-preserving flow, and let $n$ be the number of its ergodic components. Write $[\mathcal F]_1$ for its $\mathrm{L}^{1}$…

Let $(\Delta,\mathcal{U}(\boldsymbol F))$ be a right-angled building with universal group determined by local groups $F_i$, indexed by $i\in I$. Assume that each local group $F_i$…

Let $\mathbb{G}$ be a Polish group. A nonempty subset $C\subseteq\mathbb{G}$ is anti-GMS if it is nowhere dense and, for every sequence $\{U_n:n\in\omega\}$ of open neighborhoods o…

Let $\mathbb{G}$ be a Polish group. A set $S\subseteq\mathbb{G}$ is of strong measure zero, and a set $M\subseteq\mathbb{G}$ is meager. Galvin–Mycielski–Solovay characterization co…

Modulus-of-continuity conjecture. Every group topology on $G$ strictly coarser than $\tau_{st}$ is of the form $\tau_f$ for some $\mathcal{R}$-modulus of continuity.

Maximality conjecture. The $\mathrm L^1$ metric on the $\mathrm L^1$ full group of $\Phi$ is maximal.

Let $X$ be a Hausdorff topological group. Baire–banalytic characterization conjecture. The group $X$ is Polish if and only if $X$ is a Baire space and a banalytic space. This conje…

Let $G(M)$ be a topological group. It is amenable when every $G(M)$-flow admits a $G(M)$-invariant Borel probability measure. Krupiński–Pillay's conjecture. If $G(M)$ is amenable,…

Generalized Borel conjecture. The cardinality of $X$ is no larger than the weight of $G$:

Let $X$ be a metrizable space, and let $F_4(X)$ denote the corresponding free topological group of length at most four. Suppose that the set of all non-isolated points of $X$ is co…

Non-isomorphism conjecture.

Pestov's conjecture. The canonical map is an isomorphism if and only if $G$ is precompact. This conjecture concerns when the two canonical dynamical objects associated with a topol…

An $c9$-categorical structure is a structure whose theory has, up to isomorphism, a unique countable model. A structure admits a CIR when it has the property called CIR in the pape…

Let $G$ be a subgroup of a compact group. A space is weakly pseudocompact if it is $G_\delta$-dense in some compact space, meaning that every non-empty $G_\delta$-subset of that co…