  Let $G$ be a simple simply-connected connected linear algebraic group over
$\mathbb{C}$. We proved a $2$-birational Torelli theorem for the moduli space
of semistable principal $G$-bundles over a smooth curve of genus $\geq 3$,
which says that if two such moduli spaces are $2$-birational then the curves
are isomorphic. We also proved a $3$-birational Torelli theorem for the moduli
space of stable symplectic parabolic bundles over a smooth curve of genus $\geq
4$.
