6 problems

Let $C$ be a smooth projective curve with underlying Riemann surface $\Sigma_g$. Let $\mathcal{H}^{\operatorname{Betti}}$ and $\mathcal{H}^{\operatorname{Dol}}$ be the graded BPS c…

Let $\mathfrak{M}_{Dol}(r,0)$ and $\mathfrak{M}_{B}(r,0)$ be the stacks of semistable Higgs bundles and semisimple representations, respectively. Let $W_\bullet$ be the weight filt…

Let $r,n$ be integers with $r>0$, not necessarily coprime, and let $\mathcal{M}_{B}(r,n)$ and $\mathcal{M}_{Dol}(r,n)$ be the Betti and Dolbeault moduli spaces. Let…

Let $\mathcal{M}_{B}(r,n)$ and $\mathcal{M}_{Dol}(r,n)$ be the Betti and Dolbeault moduli spaces, let $h$ be the Hitchin map, and let $W_\bullet$ and $P_\bullet$ denote the weight…

Let \mathcal{X}\xymatrix@1@=15pt{\ar[r]&}\Delta be a projective minimal divisorial log terminal type III degeneration of hyperkähler manifolds, let $\mathcal{X}_t$ be a smooth fi…

P=W conjecture. For any diffeomorphism $f:X\to Y$, the $P=W$ identity holds under $f^*$.