*For any given complex projective curve C and reductive group G, Simpson's Non Abelian Hodge Theorem (NAHT) establishes a diffeomorphism between:*

*(1) the de Rham moduli space of semistable G-bundles with a flat connection on C, and*

*(2) the Dolbeault moduli space of semistable Higgs G-bundles on C.*

*If we replace the complex numbers with a field of positive characteristic, then the Dolbeault and de Rham moduli spaces share a lot of similarities from the point of view of algebraic geometry. For example, both admit nontrivial proper fibrations onto affine spaces via the Hitchin fibration and the p-curvature morphism. In fact, there is a strong version of the NAHT in positive characteristic due to Chen and Zhu, which establishes an algebraic isomorphism between:*

*(1) the stack of flat G-connections on C, and*

*(2) a twisted form of the stack of Higgs G-bundles on the Frobenius twist C'.*

*The Chen-Zhu isomorphism does not preserve semistability, so the NAHT in positive characteristic does not directly descend to the level of moduli spaces.*

*In this talk, I will explain how to construct a version of the NAHT in positive characteristic that preserves semistability, hence inducing an algebraic identification of the de Rham moduli space with a twisted form of the Dolbeault moduli space. As a consequence, we obtain an isomorphism between the intersection cohomologies of both moduli spaces, and identify the decomposition theorems for the Hitchin fibration and the p-curvature morphism.*

*This talk is based on joint work in progress with Siqing Zhang.*