192 problems

Let $S(2n,\mathbb{R})$ denote the space of real symmetric $2n\times 2n$ matrices, let $J$ be the standard symplectic matrix, and let $\rho$ be the symplectic-group action on…

For every $n\in\mathbb{N}\cup\{\infty\}$ and every ergodic treeable probability-measure-preserving equivalence relation $\mathcal{R}$ of cost $n$ on a standard probability space…

Polynomial-size uniform subset conjecture. There exists an $X$-uniform subset $T\subset G$ such that

Let $\operatorname{Map}_{g,1}$ be the mapping class group of a genus-$g$ surface with one marked point, and let $\operatorname{Map}_{g,1}(-)\subset\operatorname{Map}_{g,1}$ be the…

Abelian rank conjecture. The group $G$ must be abelian of rank at most $2$.

Let $A$ be a unital Kirchberg algebra, and let $\alpha \colon G \to \operatorname{Aut}(A)$ be an action of a finite group $G$ on $A$ such that $\alpha_g$ is outer for all…

Let $G$ be a locally compact, compactly generated group. A Hilbert length function with linear growth on $G$ is a Hilbert length function whose growth is linear. Linear-growth Hilb…

Let a lattice in $SU(1,n)$ act on the boundary of complex hyperbolic space. Boundary-action rigidity question. Is the action of any lattice in $SU(1,n)$ on the boundary of complex…

Let $\rho$ be an arithmetic isometric action of a lattice in $SU(1,n)$. Arithmetic-action deformation conjecture. Any sufficiently small perturbation of $\rho$ is of the form obtai…

Let $G=G_1\times\cdots\times G_k$ be a semisimple Lie group with all factors non-compact, and let $\Lambda<H$ be a cocompact lattice, with $G<H$. A left translation action $\rho$ o…

Let $G$ be a semi-simple Lie group with no compact factors and no simple factors isomorphic to $SO(1,n)$ or $SU(1,n)$, and let $\Gamma<G$ be a lattice. A generalized quasi-affine a…

Let $\mathcal L$ be a lamination arising from a group action. A fidelity of $\mathcal L$ is a germ map into a path-connected space whose induced map on fundamental germ groupoids i…

Let $R$ be a commutative ring with unit, and let $G$ be a discrete group without $R$-torsion. The group $G$ has jump cohomology of height $k$ over $R$ if there is an integer…

Neighborly sphere-product conjecture. There is a centrally $(\lfloor d/2\rfloor+1)$-neighborly combinatorial triangulation of every product…

Let $G:=\pi_1({\mathcal P})$ be the fundamental group of a $2$-dimensional negatively curved acute polygon ${\mathcal P}$ of finite groups. Theorem asserts that when $n=2k$ is even…

Let $G$ be a finite group, and let $r(G)$ denote its rank, namely the maximum rank of an elementary abelian $p$-subgroup of $G$ over all primes $p$ dividing $|G|$. Minimal-sphere-p…

Let $G=(\mathbb Z/2)^r$ act freely on … and let $N=n_1+\dots+n_m$. Write $\pi_G:X/G\to BG$ for the classifying map of the action. Top-dimensional surjectivity conjecture. The induc…

Let $G$ be a finite group, and let $G$ act freely on … The rank $r(G)$ is the maximum of the $p$-ranks $r_p(G)$ over primes $p$ dividing $|G|$, where $r_p(G)$ is the largest $r$ su…

Let $(X,o)$ be a ${\mathbb Q}$-Gorenstein surface singularity with ${\mathbb Q}$-homology sphere link, and let its resolution graph be equipped with the associated discriminant-gro…

Rationality and integrality conjecture. For every degree $p$,

Let $k$ be a field, let $G$ be a finite group, and let $X$ be a $k$-variety with a faithful $G$-action. For every prime $p$, let $G_p$ be a Sylow $p$-subgroup of $G$. Duncan's conj…

Eventual 8g conjecture.

Characteristic-invariant conjecture. The action $\alpha$ is classified up to cocycle conjugacy by the characteristic invariant $(H,\chi)$.

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, and let $\mathcal{C}$ be a finite category. If $\mathcal{C}$ is a fin…

In the first case, let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$ and let $p$ be any prime. In the second case, let $\Gamma$ be a cocompact arithmetic lattice of the second type in…