The natural action of the multiplicative group on the Dolbeault moduli space of Higgs bundles extends to an action on the Hodge moduli space of $\lambda$-flat bundles. When restricted to a general fiber, this action induces a stratification of the de Rham moduli space of flat bundles. In this talk, we will discuss some conjectures proposed by Simpson concerning the geometry of this stratification, together with recent progress toward resolving them. This is based on joint work with Z. Hu and R. Zong.
