18 problems

Logarithmic Fourier-norm conjecture. If, for every finite subgroup $V\leq\widehat G$ with $|V|\leq M$, one has

Generalized Duistermaat–Heckman conjecture. An analogous correlation-function formula should hold for every compact group, with the triangular integration domain identified with…

Let $G$ be a compact group with Haar probability measure $\mu$, let $\rho\colon\Gamma\to G$ be a homomorphism, and let $\Gamma$ act on $(G,\mu)$ by left translation through $\rho$.…

Let $G$ be a compact group with Haar probability measure $\mu$, let $\rho\colon\Gamma\to G$ be a homomorphism, and let $\Gamma$ act on $(G,\mu)$ by $\gamma g=\rho(\gamma)g$. For ea…

Let $G$ be a compact Hausdorff topological group, and define its Witten zeta function by … where the sum runs over the finite-dimensional irreducible representations of $G$. Kuroka…

Let $G$ be one of $\mathrm{SU}(n)$, $\mathrm{SO}(n)$, $\mathrm{Spin}(n)$, or $\mathrm{Sp}(n)$, and let $\mathcal{A}\subseteq G$ be measurable with measure $\nu$. The diameter with…

Let $G$ be a compact group equipped with its Haar probability measure, and let a Kedlaya set be a set of the form … where $G$ acts on a set $X$, $B\subseteq X$, and $x\in X$. A mea…

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…

Bell–Miles–Ward's conjecture. The function $\tilde{\zeta}_{\theta}(z)$ is either rational or admits a natural boundary. This is a Pólya–Carlson-type dichotomy for a dynamical zeta…

Let $G=\{G(N)\}_N$ be a natural family of groups, and let $\chi=\{\chi_N\}_{N\geq N_0}$ be a stable irreducible character with $\chi_N\in\widehat{G(N)}$. Let $\pi(w)$ denote the pr…

Let $G$ be a profinite group, and let $n$ be a positive integer. Suppose that the set of solutions of … has positive Haar measure. Lévai–Pyber conjecture. Then $G$ has an open subg…

Let $\mathbf{F}$ be the free group, and let $w_{1},w_{2}\in\mathbf{F}$. For a compact group $G$, write $\mu_{w}$ for the probability measure on $G$ obtained by evaluating $w$ on a…

Let $\mathbf{F}_r$ be the free group of rank $r$, and let $w\in\mathbf{F}_r$ induce a probability measure on each compact group via its word map. Word-measure rigidity conjecture.…

Griesmer's conjecture. Under these hypotheses, such compact subsets $A'$ and $B'$ exist. This conjecture proposes an inverse theorem for near equality in Kneser's inequality that e…

Let $\cal F$ be a Riemannian foliation with bounded holonomy, and suppose that no leaf has infinite fundamental group. Let $H_p$ denote the holonomy group at $p$. Bounded-holonomy…

Commutativity conjecture. If $G$ is a pseudofinite group, then $(bG)^{0}$ is commutative.

Quantum-dimension conjecture. For every $\chi\in\operatorname{Irr}(G)$,

Let $f$ be a finitely $\mathcal{A}$-determined multigerm which is also a critical normalization. Write $D(f)=f(\Sigma(f))$ for its discriminant, and let…