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…

Let $p$ be a prime, let $G=({\Bbb Z}/p)^k$, and let $X$ be a finite free $G$-complex. Carlsson's toral rank conjecture. One has … This conjecture gives a lower bound on the total m…

Koopman's irreducibility conjecture. The representation $\pi^{\alpha,\mu,X}$ is irreducible if and only if

Kropholler's conjecture. If there is a proper $H$-almost invariant set $A$ such that $A=AH$, then $G$ has a non-trivial action on a tree in which $H$ fixes a vertex $v$ and every e…

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

Let $q$ be a power of $3$, let $G=PGL_2(q)$ act on the lines of $PG(3,q)$, and let $C$ be the twisted cubic. A line is generic if it neither intersects $C$ nor lies in any osculati…

Let $G$ be a compact group acting continuously and transitively on a compact Hausdorff space $X$. A space of continuous functions on $X$ is called minimal if it is closed, invarian…

Borovik–Cherlin's conjecture. The group $V$ has the structure of an $n$-dimensional vector space over an algebraically closed field $F$ of Morley rank $1$, and

Brown's conjecture.

Let $p$ be a prime, let $n_1,\ldots,n_k$ be positive integers, and let $(\mathbb{Z}_p)^r$ act freely on the product $S^{n_1}\times\cdots\times S^{n_k}$. Free-rank conjecture for pr…

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…