104 problems

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…

Let $G$ be a locally compact group. Write $\operatorname{TLIM}(G)$ for the topological left-invariant means on $L_\infty(G)$ and $\operatorname{TIM}(G)$ for the two-sided topologic…

Let $X$ be a metrizable space, and let $F_4(X)$ denote the subspace of the free topological group $F(X)$ consisting of words of reduced length at most $4$ with respect to the free…

Block decomposition conjecture. There should be a continuous function $u$ from $\Pi_n$ into $U(n)$ such that each vertex is mapped to the corresponding permutation matrix and the b…

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 $U({\mathcal H})_u$…

Admissibility conjecture. The assumption that $G$ satisfies the two-three condition is only technical and is forced by the techniques used to prove admissibility of compact groups.

Admissibility conjecture. It should be possible to remove the two-three condition using other techniques.

A space metrizable by a complete metric is considered, together with zero-dimensional metrizable groups and the property of being strongly zero-dimensional. Complete-metric embeddi…

Let $G$ be a separable, metrizable, connected topological group of finite dimension, and suppose that $G$ is complete. An arc component of the identity is the subset of points that…

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…

Let $G$ be a locally compact abelian group with separable dual group $\hat{G}$, and let $\varphi\in\mathcal{H}(G)$ be a Hartman measurable function. Write…

Let $\mathrm{G}(\mathbb{Q})$ be an algebraic group admitting an embedding of $\mathbb{Q}$, and let $\mathrm{G}(\mathbb{Q}_p)$ and $\mathrm{G}(\mathbb{R})$ denote its groups of poin…

Let $\mathbb{Z}(p^\infty)$ be the Prüfer group, and consider the construction of a non-minimally almost-periodic, non-maximally almost-periodic Hausdorff group topology on…

Let $P$ be the pseudoarc, the unique hereditarily indecomposable chainable continuum, and let $G=H(P)$ be its homeomorphism group. Let $M_G$ denote the universal minimal compact…

Dimension bound conjecture. The dimension of a topological basis of a topologically free subgroup of a free abelian topological group $A(X)$ cannot exceed $\dim X$:

For a group code $\mathbb{{\mathcal{C}}}$ with symbol groups $G_k$ and restrictions $\mathbb{{\mathcal{C}}}_{|\{k\}}$ at each time $k$, equipped with the appropriate topologies, Du…

Let $G=\operatorname{Homeo}(P)$ be the homeomorphism group of the pseudo-arc $P$, acting on $P$ by evaluation. A universal minimal flow of $G$ is a minimal compact Hausdorff $G$-fl…

Topology comparison conjecture. Under mild conditions, $\pi_n^{\tau}(X,x_0)$ is finer than $\pi_n^{pSpan}(X,x_0)$.

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 $G$ be a compact topological group, and let … be its density spectrum of dense subgroups. Peng's conjecture. Every compact group $G$ satisfies … This asserts that every compact…

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…

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

Semiproximality conjecture. An $$ -embeddable almost disjoint family is semiproximal if and only if the range of some (equivalently, every) embedding contains no uncountable perfec…

Finite-singularity direct-limit conjecture.