50 problems

Guichard-Labourie-Wienhard conjecture. The set of $\Theta$-positive representations $\rho:\pi_1(\Sigma_g)\to G$ is open and closed in $\operatorname{Hom}(\pi_1(S),G)/G$. In particu…

Let $\Gamma$ be a discrete linear group with Kazhdan's property (T). For each integer $n\geq 1$, let $\varkappa_n(\Gamma)$ denote the dimension-growth invariant used in the paper f…

Let $K$ be a compact connected Lie group, let $c$ be a central value, and let $\mathcal{R}^{g}_{K}(c)$ denote the corresponding moduli or representation space for a genus-$g$ surfa…

Let $\Gamma\subset\operatorname{Isom}(\mathbb H^n)$ be a lattice, let $\Gamma'\subset\Gamma$ be a finite-index subgroup, and let $\rho$ denote the relevant natural representation.…

Let $V_n(A)$ denote the scheme parametrizing $n$-dimensional representations of an algebra $A$. For the polynomial ring in two variables over the complex numbers, consider…

Neumann's surjectivity conjecture. If $k=1$, then $\mathbb{U}$ is onto. In general, every $\mathsf{PSL}(2,\mathbb{R})$-representation with dense image lies in…

Euler-component ergodicity conjecture. For each integer $1\leq k\leq 2g+b-2$, the $\mathsf{Mod}_{\Sigma}$-action on the component $e^{-1}(2-2g+b+k)$ of $\mathsf{Hom}(\pi,G)$ is erg…

Dynamical zeta convergence conjecture. There exists an open set with non-empty interior in each component of $\operatorname{Rep}(\Gamma;V)$ on which this formal limit has true conv…

Isomorphism conjecture. The map $\Phi$ is an isomorphism of algebraic varieties. This would identify the moduli of the relevant irreducible representations with the explicitly defi…

Let $O$ be the base orbifold of the Seifert-fibered homology $3$-sphere under consideration, and let $\mathcal{R}^+_{\rm{U}(2,1)}(O)$ denote the relevant representation space. The…

Let $X$ be a homology 3-sphere, and let $\tau_{SU(3)}(X)$ denote its SU(3) Casson invariant. Evenness conjecture. The invariant $\tau_{SU(3)}(X)$ is even for every homology 3-spher…

Let $\Sigma$ be a closed compact orientable surface of genus $l>1$, and let $G$ be a connected complex semisimple Lie group. Write $\pi_0(\operatorname{Hom}(\pi_1(\Sigma),G))$ for…

Let $\Sigma$ be a punctured surface, and let $\mathcal R(\Sigma)$ denote the space of type-preserving representations $\phi:\pi_1(\Sigma)\to\mathrm{PSL}(2,\mathbb{R})$. For a relat…

For a positive integer $n$, let $C_n(\operatorname{GL}_r(k))$ denote the variety of commuting $n$-tuples in $\operatorname{GL}_r(k)$. Motivic stability conjecture. The limit … exis…

For $n\geqslant 2$, let $\Gamma$ be a cocompact lattice of $\operatorname{O}(n,1)$, and let $N\geqslant n$. Consider the standard embedding … A representation is strictly dominated…

Nonexistence conjecture. There exist no $\mathbb Z_2$-harmonic $1$-forms on $Y$ with $\mathcal Z\neq\emptyset$ with respect to any metric. In particular, there exist no…

Component-count conjecture. There are at most

Let $H_n$ be a twisted Hopf link, and let $X(H_n,G)$ denote its $G$-character variety. For an integer $r\geq 1$, consider the character varieties for $G=\operatorname{SU}(r)$ and…

Let $p>2$ and consider the smooth connected components of the space $\textnormal{Pos}_{p,p+1}(\Gamma)$ of $\Theta$-positive representations of a surface group $\Gamma$ into…

Parabolic-representation existence conjecture. A knot $K$ is nontrivial if and only if $8par(K)$ is nonempty. Moreover, $8par(K)$ has a zero-dimensional component.

Let $S$ be a closed orientable surface and let $G$ be a simple Lie group. Consider components of the representation variety of surface groups into $G$ that consist entirely of disc…

Let $\mathcal{X}$ be a reduced $\mathbb{Z}$-scheme and let $X=\mathcal{X}(\mathbb{C})$ be the associated complex algebraic variety. Say that $X$ is strongly polynomial count when i…

A complex variety $X$ is a variety over $\mathbb{C}$. It is asymptotically polynomial count when its point counts over finite fields agree with a polynomial in the field size for a…

Let $\Gamma$ be a finitely generated supersolvable group, meaning that it admits an invariant normal series whose factors are cyclic. A group is flawed if, for every reductive…

Let $r\geq 1$ and let $n_1,\ldots,n_r$ be positive integers. A generalized torus knot group is a group with presentation … When $r=2$, these are torus knot groups. Generalized toru…