80 problems

Let $K$ be a Hilbertian field of characteristic $0$, and let $n\geq 2$. For a polynomial $f\in K[x]$, write $f^{-k}(0)$ for the set of $k$-th preimages of $0$, and let $[S_n]^k$ de…

Let $G=\mathbb{Z}/2\mathbb{Z}\wr\mathbb{Z}^d$ with $d\geq 1$, and let $\mu$ be a non-degenerate finitely supported measure on $G$ whose projection to $\mathbb{Z}^d$ induces a trans…

Sagan–Swanson's conjecture. The set of monomials ${\mathcal{A}}_{n,r}$ descends to a vector space basis of $SR_{n,r}$.

Let $\Gamma$ be a finite subgroup acting on $\mathbb C^2$, let $\Gamma_n$ be its wreath product acting on $\mathbb C^{2n}$, and let … be the natural morphism over…

Alspach et al.'s wreath-product conjecture. If $G$ and $H$ are hamiltonian decomposable directed graphs, then $G\wr H$ is also hamiltonian decomposable.

Cornulier–Stalder–Valette's conjecture. The Haagerup property for $G$ and $H$ should imply the Haagerup property for every permutational wreath product $G\wr_X H$.

Let $\Gamma$ be a finite subgroup of $SL_2(\mathbb{C})$, let $\Gamma_n=\Gamma^n\rtimes S_n$, and let $X_{\Gamma,n}$ be the union of connected components of the $\Gamma$-fixed locus…

Let $V$ be a ${\mathscr{A}}_{n,\lambda,\nu}$-module, and let $F_i$ be the functor defined in the paper. Write $H^r({\mathscr{C}}^\bullet(V))$ for the $r$-th cohomology of the compl…

Let ${\Gamma}$ and $G$ be finite groups. For each $n$, consider the complexified rings $\mathbb C\otimes_{\mathbb Z}\mathcal G_{\Gamma}(n)$ and…

Let $W={\mathfrak S}_n\ltimes({\mathbb Z}/N{\mathbb Z})^n$ act naturally on ${\mathfrak h}={\mathbb C}^n$, and let $p:{\mathcal C}_W\to{\mathfrak h}/W\times{\mathfrak h}/W$ be the…

Wreath-product orbifold elliptic genus conjecture. The elliptic genus for the wreath product orbifold $Y^n/G_n$ satisfies

Proposed colored-face model. The ideal $SI'_{n,r}$ is stable under ${\mathbb{Z}}_r\wr{\mathfrak S}_n$, and there is a module isomorphism

Murai–Rhoades–Wilson's conjecture. For every $d\geq 0$, there exists a $G$-equivariant surjection

The LEF nilpotent-product conjecture. For each $k\geq 1$, the $k$-nilpotent product of a finite family of LEF groups is LEF.

Let $G$ be a finitely presented self-similar metabelian group, and let $\mathbb{Z} \wr \mathbb{Z}$ denote the wreath product of two copies of the infinite cyclic group. The authors…

Let $G$ and $F$ be groups, and define their wreath product by $G\wr F=(\prod_F G)\rtimes F$, with $F$ permuting the factors. The Full Farrell–Jones conjecture. A group $G$ satisfie…

Coamenable control conjecture. Some control on the linear estimates guaranteed for fixed countably infinite amenable $A$ in the cited work for wreath products may be obtained throu…

Let $L(\sigma)$ denote the statistic defined for the wreath product action, with $\sigma$ sampled from the relevant wreath product of parameter $k=2$. Wreath product longest increa…

Wreath Macdonald norm conjecture. The wreath Macdonald polynomials satisfy

Let $G$ be a finite group, let $F_r$ be the free group of rank $r$, and let $w\in F_r$. For a stable irreducible character $\chi=(\chi_n)_{n\in\mathbb{N}}$ of $G\wr S_{\bullet}$, w…

Chen–Proudfoot's conjecture. If $V$ is $G$-equivariantly log-concave, then $V^{\otimes n}$ is $W$-equivariantly log-concave.

Let $\Sigma_n\curvearrowright X_n$ be an asymptotically oligomorphic sequence of group actions, let $G$ be a compact group, and let $\phi\in\widehat G\setminus\\{\mathbf{1}\\}$. Fo…

Let $W_n$ be the ambient binary wreath-product group acting on $\{1,\dots,2^n\}$. For $n\geq 1$, let $H\leq W_n$ be a subgroup containing a permutation $\sigma\in W_n$ that acts tr…

Let $\Delta$ be a finite group. The wreath product $\Delta \wr \mathbb{Z}$ is locally stable. Finite-wreath-product local stability conjecture. Every finite wreath product of this…

Let $K/H$ be the set of equivalence classes of $K$ under the action of $H$, and let $N$ denote the length of a surjective strategy. Orbit-bound conjecture. If $G\wr H$ has a surjec…