12 problems

The P=W conjecture. Under this identification, the perverse filtration on $M_D$ associated with the Hitchin fibration matches the Hodge-theoretic weight filtration on $M_B$. This c…

Let $G=\operatorname{SL}_n$, let $\hat{G}=\operatorname{PGL}_n$, and let $\Gamma=\operatorname{Pic}^0(C)[n]$. Fix a line bundle $D$ on $C$ that is either isomorphic to the canonica…

Let $n,d_1,d_2$ be integers satisfying … For each $d$, let $h_{n,d}$ denote the Hitchin fibration on the corresponding Higgs-bundle moduli space, and let…

Let $\mathfrak{g}_{\mathbb{R}}$ be a real form, let $\hat{\mathcal{T}}_{\mathfrak{g}_{\mathbb{R}}}$ be the moduli space of $\mathfrak{g}_{\mathbb{R}}$-complex structures, and let t…

Let $C$ be an irreducible nonsingular projective curve of genus $g\geq 2$, let ${\mathcal M}_{\mathrm{Dol}}$ be its twisted Dolbeault moduli space with Hitchin fibration $h$, and l…

The P=W conjecture in Hodge–Tate form. The conjecture predicts

Numerical $P=W$ conjecture. The perverse numbers of the Hitchin fibration and the Hodge numbers of the corresponding character variety should agree:

Let $M_B$ be the Betti moduli space, or character variety, of local systems, equipped with the mixed Hodge structure on its cohomology. Let $M_{Dol}$ be the corresponding Dolbeault…

Let ${\rm M}_{Dol}(S,C_{\cdot})$ be the Dolbeault (Hitchin) moduli space and ${\rm M}_B(S,C_{\cdot})$ the Betti character variety. Let $N^{\ast}_{Dol}$ and $N^{\ast}_B$ be neighbor…

Let $\mathcal{R}$ be a character variety and let $\mathcal{M}$ be the corresponding moduli space of semistable Higgs bundles on a compact curve. Non-abelian Hodge theory induces an…

Let $\mathcal{R}_{g,\boldsymbol{\mu}}$ be the character variety, let $\overline{\mathcal{R}_{g,\boldsymbol{\mu}}}$ be the compactification constructed from the representation varie…

ORS geometric conjecture. The reduced HOMFLY homology of the knot link$(C,x)$ has Poincare polynomial