45 problems

Let $F$ be a free group, and associate to each element of $F$ the feature vector of weights of its Whitehead Graph in the corresponding feature space. A region is a bounded subset…

Let $F$ be a free group. For each positive integer $n$, let $U_n$ be the set of all non-minimal elements in $F$ of length $n$, and let $NU_n\subset U_n$ be the subset consisting of…

For each $n,N\geq 1$, let ${\underline{\operatorname{Aut}}}^{\leq N}(A_{n,\mathbb Q})$ and ${\underline{\operatorname{Aut}}}^{\leq N}(P_{n,\mathbb Q})$ denote the filtered automorp…

Tameness Conjecture. The automorphism $f$ can be isotoped so that it preserves the splitting and the induced automorphism $g$ on $H_1$ is train-track generic and has a tame one-dim…

Let $\overrightarrow{u}$ be an autocorrect sequence, and let a $\overrightarrow{u}$-Engel automorphism mean an automorphism satisfying the corresponding Engel condition. Let $G$ be…

Let $\overrightarrow{u}$ be an autocorrect sequence, and let a $\overrightarrow{u}$-Engel automorphism mean an automorphism satisfying the corresponding Engel condition. Consider a…

Let $X^{an}/K$ be an analytic K3 surface with the lattice $\Lambda_X$, period homomorphism $\rho_X$, valuation vector $v_X:=val_K\circ\rho_X$, and affine piecewise-linear action on…

Medvedev's conjecture. $L$ contains a solvable subring of $m$-bounded derived length whose index is $(p,n,m)$-bounded.

Khukhro's conjecture. $G$ contains a subgroup of $m$-bounded derived length and $(p,n,m)$-bounded index.

Khukhro's conjecture. $G$ contains a subgroup of $m$-bounded nilpotency class and $(p,m)$-bounded index.

Let $X$ be a normal projective variety of dimension $d\ge 2$ over $\mathbf{k}$, and let $f$ be an automorphism of $X$ such that $\deg_1(f^n) \asymp n^{k}$ as $n\to\infty$. Lower-bo…

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, and let $\mathcal{C}$ be a finite category. Let $\operatorname{Aut}(\mathcal{C})$ denote its automorphism gro…

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma$ be the Galois group of $k/\mathbb{F}_p$, and let $G$ be a finite group. Let…

Let $S$ be a smooth projective surface, and let $G\subset\mathrm{Aut}(S)$ be a group of automorphisms. Assume that every element of $G$ acts trivially on $H^{2,0}(S)$. Trivial-acti…

Let $Y$ be a Calabi–Yau manifold, meaning that $Y$ has trivial canonical bundle and $h^1(\mathcal{O}_Y)=0$. Let $\mathrm{Nef}^+(Y)$ denote the convex hull of the rational points of…

Let $k$ be a field of characteristic $p>0$, let $E$ be a supersingular elliptic curve over $k$, and for $g\geq 2$ let $\mathcal{A}_g$ denote the moduli space of principally polariz…

Let $M$ be a type $\rm III_1$ factor with separable predual. An automorphism $\theta\in\operatorname{Aut}(M)$ is pointwise inner if, for every positive linear functional…

Let $G$ be a reductive group, let $\Out_F(G)$ be its group of $F$-rational outer automorphisms, let $\theta\in\Out_F(G)$, and let $\mathfrak w$ be a Whittaker datum. Outer-automorp…

Let $X$ be a smooth projective K3 surface, and let ${\rm Aut}^s(X)$ denote its group of symplectic automorphisms. For $\phi\in{\rm Aut}^s(X)$, write … for the induced morphism. Blo…

Let $F_n$ be the free group of rank $n$, and let $f,g$ be automorphisms of $F_n$. Write $\operatorname{deg}(f)$ and $\operatorname{deg}(g)$ for their degrees. Degree-composition co…

Let $X$ be a compact Kähler space. An automorphism $\sigma\in\operatorname{Aut}(X)$ is wild if every non-empty analytic subset $Z\subseteq X$ satisfying $\sigma(Z)=Z$ is equal to…

Let $M$ be a type ${\rm III}_1$ factor with separable predual. An automorphism $\Theta\in\operatorname{Aut}(M)$ is pointwise inner if, for every normal state $\psi\in M_*$, there i…

Uniform Fitting-height conjecture. For every $f(x)\in\mathbb{Z}[x]\setminus\{0\}$, there exist integers $k(f(x))>0$ and $m(f(x))$ such that, whenever $G$ is a finite solvable group…

Let $X$ be an algebraic variety over an algebraically closed field of characteristic zero, and let $\phi:X \longrightarrow X$ be an endomorphism. Let $Y$ be a proper closed subvari…

Solvable-group supremum conjecture. For every positive integer $n$, one has