11 problems

Let $C$ be a rational integral projective curve with a unique irreducible planar curve singularity $(C,0)$, let $\overline{J}C$ be its compactified Jacobian, and let…

Let $S$ be the surface and let $Coh^{ss,0}_c(S)$ denote the category of parabolic semistable coherent sheaves on $S$ of parabolic degree zero with pure one-dimensional compact supp…

Let $\Sigma$ be a projective curve of genus $g\geq 2$, and let $n,d$ be coprime integers. Let $M_{\mathrm{Dol}}$ be the moduli space of stable Higgs bundles on $\Sigma$, equipped w…

Cohomological uniqueness conjecture. The solution to Equation $$ is unique in cohomology, including outside the redundancy range.

Positivity conjecture. In the redundancy range,

Enumerative P=W conjecture. In the redundancy range, there exists a unique defect-homogeneous $F\in H^*(Z)$ with $\operatorname{def}(F)=2k-2$ such that, for every defect-homogeneou…

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

Let $M_B$ and $M_{\mathrm{Dol}}$ be the Betti and Dolbeault moduli spaces, respectively. Let $N^*_B$ and $N^*_{\mathrm{Dol}}$ be punctured neighbourhoods at infinity, let…

Let ${\mathcal{M}}_\Dol^{\bm \mu}$ be the parabolic Higgs moduli space and ${\mathcal{M}}_\B^{\bm \mu}$ the corresponding character variety, related by non-abelian Hodge theory. Th…

Let $C$ be a complex smooth projective curve, let $D=p_1+cdocdots+p_k+rp$, and let ${\mathcal{M}}_\Dol^{{\bm \mu},r}$ be the corresponding moduli space of stable twisted parabolic…