14 problems

Standard-action conjecture. Every quasi-action by $\Gamma$ on a Gromov hyperbolic metric space either has an invariant horoball or is standard.

Finiteness conjecture. If $\Gamma$ is an irreducible lattice in a higher rank Lie group (or in a nontrivial product of locally compact groups) $G$, there are finitely many Gromov h…

Let $n$ be the dimension parameter, let $\rho=(n-1)/2$, and define … where $F(a,b;c;z)$ is the hypergeometric function and $k$ is complex. The zero-free region conjecture. For ever…

Let $n$ be the dimension parameter, let $\rho=(n-1)/2$, and let $F(a,b;c;z)$ denote the hypergeometric function. For the function $F_k(z)=F(k,-\rho;k+n/2;z)$, with complex variable…

Pucci–Serrin's conjecture. The existence of a solution requires $\lambda$ to be larger than some positive constant; equivalently, there is a constant $\underline{\Lambda}>0$ such t…

Strong Property (T) action conjecture. Any action of $G$ by isometries on $X$ is elementary.

Let $k$ be a positive integer, let $n\in\{2k+1,2k+2,\ldots,4k-1\}$, and consider the higher-order Brézis–Nirenberg problem on a Euclidean ball $\Omega=B^n$: … where…

Cornulier's conjecture. There exist negatively curved homogeneous spaces other than symmetric spaces that are quasi-isometric to finitely generated groups.

Let $A_\Gamma$ be an Artin group, and let $\mathcal{C}(A_\Gamma)$ denote its graph of irreducible proper parabolic subgroups of finite type. The graph is considered for Artin group…

Let $X$ satisfy the hypotheses of Theorem K-detection, so that the theorem concludes that $X$ is $p$-hyperbolic. Here, $p$-hyperbolicity and $\mathbb{Z}/p^r$-hyperbolicity refer to…

Let $A_S$ be an irreducible non-cyclic FC type Artin group, and let ${\mathcal{CP}_{A_S}}$ denote its complex of parabolic subgroups. Hyperbolicity and infinite-diameter conjecture…

Let $u\in E_{\lambda}$ with $\lambda\in\mathbb R^\times$, let $p\in(1,2)$, and let $X$ be a hyperbolic space over $\mathbb R$, $\mathbb C$, or $\mathbb H$. The notation…

Let $H^n$ be the $n$-dimensional hyperbolic space, and let $\Omega_*$ be the potential obtained from the wrapping construction for this symmetric space. Consider the solution to th…

Fix a surface $S$ and a complexity parameter $xi$ with $\xi>0$. Let $r_\xi(S)$ be the maximum number of pairwise disjoint subsurfaces of complexity $\xi+1$ embedded in $S$, and let…