24 problems

Polynomiality and positivity conjectures. The following two assertions are conjectured:

Hausel's mixed-Hodge polynomial conjecture.

Genus-one partition identity. The following combinatorial identity holds:

Fix the number $k$ of marked points and the genus $g$. For partitions $\mu_1,\ldots,\mu_k$ and the corresponding generic character variety $X_{g;J_{\mu_1},\ldots,J_{\mu_k}}$, let…

Mixed-Hodge-polynomial conjecture. The total number of fixed manifolds of $\mathcal{M}_{g,n}$, equivalently the number of simple modules of $V_{g,n}$, is

Let $C_{g,n}$ be the character variety of the $SU(2)/\mathbb{Z}_2$ Hitchin system on the punctured surface, and let $PH_{g,n}(q)$ be its pure mixed Hodge polynomial. Define the ren…

Libgober's conjecture. If $X$ is rationally elliptic, then its mixed Hodge structure is of Hodge–Tate type; equivalently, every non-zero mixed Hodge number $h^{k,p,q}$ satisfies

Let $\mathcal{C}$ be a $k$-tuple of semisimple orbits, let $\alpha_{\mathcal{C}}$ be its associated dimension vector, and let…

Let $\mathcal{C}$ be a generic $k$-tuple of semisimple conjugacy classes, let $\alpha_{\mathcal{C}}$ be its associated dimension vector, and let…

Let $\mathcal{M}_{\mu,\nu}$ be the partial resolution of the character variety defined in the source, and let $w$ be an $n$-cycle in the Weyl group relative to the third puncture.…

Let $\mathcal{C}=(C_j)_{1\le j\le 2k}$ be a strongly generic tuple of semisimple conjugacy classes in $\operatorname{GL}_n\langle\sigma\rangle$ satisfying the condition referred to…

Let $\mathcal{C}$ be a generic tuple of conjugacy classes, let $\bs\mu$ encode the eigenvalue multiplicities, and let $\mathcal{M}_{\mathcal{C}}$ be the associated character variet…

Let $U$ be a log Calabi–Yau variety and assume that its homological mirror $U^\vee$ is also a log Calabi–Yau variety with the same dimension. Set $d=\dim U=\dim U^\vee$. The perver…

For a partition $\lambda$, let $\mathcal{H}_{\lambda}(z,w)$ be the genus-one hook function, and let $lvertlambdarvert$ denote the size of $\lambda$. Hausel–Rodriguez-Villegas genus-on…

Let $\mathcal{M}_n$ be the moduli space under consideration, let $d_n$ denote its dimension, and let $\mathcal{H}_{\lambda}(z,w)$, $\overline{H}_n(z,w)$, and $H(\mathcal{M}_n;q,t)$…

Let ${\mathcal{M}}_\B^{{\bm \mu},{\bf r}}$ be the generic wild character variety, let $d_{{\bm \mu},{\bf r}}$ be its dimension, and let…

Let ${\mathcal{M}}_\B^{{\bm \mu},{\bf r}}$ be the generic wild character variety, let $d_{{\bm \mu},{\bf r}}$ be its dimension, and let…

Let $X$ be a complex analytic space that is $S_2$ and has pure dimension $\geq 3$, let $x\in X$ be a point, and let $\pi:\bar X\to X$ be the normalization with corresponding point…

Let ${\bf C}$ be a generic tuple of conjugacy classes of $\operatorname{GL}_n(\mathbb{C})$ of type ${\bm\omega}\in\overline{{\mathbf{T}}}^{o}$. Let $\iota^k({\bm\omega})$ be the ty…

Let ${\bf C}$ be a generic tuple of conjugacy classes of $\operatorname{GL}_n(\mathbb{C})$, with type ${\bm\omega}\in\overline{{\mathbf{T}}}^{o}$, and let $d_{\bf C}$ denote the di…

For a partition $\lambda$ of $n$, let $C_\lambda$ be the unipotent conjugacy class of $\operatorname{GL}_n(\mathbb{C})$ whose Jordan-block sizes are given by the dual partition…

Let $(Y,y)$ be a singularity, and let $w_{j}^i(y)$, $\bar{w}_{j}^i(y)$, and $\ell_{j}^i(y)$ denote the invariants defined from the mixed Hodge structures associated with the singul…

Let ${\bm \mu}=(\mu^1,\ldots,\mu^k)\in{\mathcal{P}_n}^k$ be a tuple of partitions, let ${\mathcal{M}}_{\bm \mu}$ be the corresponding $\operatorname{GL}_n(\mathbb{C})$ character va…

Let $\M^\mu_\B$ be the smooth character variety associated with generic semisimple conjugacy classes of parabolic type $\mu$, let $\bar{\M}^\mu_\B$ be its corresponding $\PGL_n$ qu…