26 problems

Torus-knot conjecture. If $K$ is a nontrivial $SU(2)$-averse knot, then $K$ is a torus knot.

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.…

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…

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…

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…

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…

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 $\mathcal{X}$ 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…

Guichard–Wienhard conjecture. The extra components in the moduli space of $SO(p+q,q)$-Higgs bundles conjectured to contain positive representations are those connected components c…

Let $M$ be a taut sutured genus-two handlebody. A representation $\alpha:pi_1(M)\to\operatorname{GL}_1(\mathbb C)$ is certifying when the inclusion-induced maps … are isomorphisms.…

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…

Guichard-Wienhard equivalence. There are connected components of $\operatorname{Hom}(\pi_1(S),G)/G$ which are not distinguished by characteristic invariants if and only if there ex…

Given a knot $K$ in the 3-sphere, let … where $\mu$ is a chosen meridian and … For a trefoil $T$, its representation variety is $\mathscr{R}(T,\mathbf{i})\cong\{*\}\sqcup S^1$, wit…

Let $G$ be a complex algebraic group, and let $\mathrm{K}\operatorname{M}_{G}$ denote the ambient module of motivic data associated with $G$. Define … the submodule generated by th…

Let $\Gamma_o$ be the group of the six-point configuration and let $X_0$ denote the relevant character space of representations into $\mathrm{PSL}_2(\mathbb{R})$. For a representat…

Modified Lee–Zhang conjecture.

Let $F_r$ be a free group, and call two pairs of words $(u,v)$ and $(u',v')$ $\operatorname{SL}_n$-trace equivalent when the corresponding words have equal traces under every repre…

Goldman's volume rigidity conjecture. Equality holds if and only if $\rho$ is a discrete, faithful representation of $\Gamma$ into $G$. Goldman proved this conjecture for all conne…

Redundant-representation conjecture. (a) The action of $\operatorname{Out}(F_n)$ on $R$ is ergodic.

Twisted diagonal deformation conjecture. If $G$ is not locally isomorphic to $\mathrm{Sp}(2n,\mathbf{R})$, then every maximal representation

Topological invariants conjecture. If $G$ is not locally isomorphic to $\mathrm{Sp}(2n,\mathbf{R})$, then the topological invariants of Theorem distinguish connected components of…