45 problems

Let $G$ be a connected compact simple real linear algebraic group, and let a word map be the map induced by a word in finitely many variables on $G$. Larsen's conjecture. Every wor…

Let $n\geq 2$ and consider $n$-tuples in the special unitary group $\mathrm{SU}(2)$. The subgroup generated by an $n$-tuple acts naturally on $L^2(\mathrm{SU}(2))$. Gamburd–Jakobso…

A homogeneous space of a compact Lie group is a quotient equipped with the induced smooth structure, and its dimension as a differentiable manifold is its usual manifold dimension.…

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 a pseudofinite group be a group satisfying the theory of all finite groups, and let its Bohr compactification be its universal compactification, namely a compact group receivin…

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

Fourier-norm conjecture. If $|\alpha-1/3|\leq\epsilon$, then

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 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 $G$ 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…

For $\mathrm{SU}(n)$, let the maximal Haar measure of a measurable product-free subset be the supremum of the Haar measures of measurable subsets $A\subseteq \mathrm{SU}(n)$ satisf…

Let $G$ be a compact, semisimple group and let $A$ be a finite subset that densely generates $G$. A uniform spectral gap means that the operator averages associated with $A$ have a…

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…

Spectral gap conjecture. If $\nu^{*n}\to \mathrm{Vol}$ weakly as $n\to\infty$, then $\nu$ has a spectral gap. Equivalently, in the nonuniform spectral gap estimate, the logarithmic…

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 compact group, let $\nu$ be an adapted probability distribution on $G$, and let $\pi$ be a nontrivial irreducible unitary representation of $G$. Define the character…

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 $s eq 1$? No: the source considers $1leq s<infty$. Let $gamma^{s}{mathbf{X},L^2}(SU(2))$ and $gamma^{(s)}{mathbf{X},L^2}(SU(2))$ be the corresponding sub-Laplacian Gevrey space…

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 $G$ be a group, and say that $G$ has the strong Borsuk–Ulam property when every relevant equivariant mapping problem satisfies the strong Borsuk–Ulam conclusion described in th…