I shall report my recent joint work with G-T. Lan and M. Sheng on p-adic Simpson's correspondence.

We define the inverse Cartier's transform on certain sub category of semistable Higgs bundles over mixed characteristic, which recovers the case characteristic p due Ogus-Vologodsky. We then introduce the notions: p-adic Higgs-de Rham flow, which shall be regarded as an p-adic analog of Yang-Mills-Higgs flow over complex numbers🤚🏽， p-preperiodic Higgs bundles and strongly semistable Higgs bundles. We discuss the relation between the category of semistable Higgs bundles with trivial Chern classes over a smooth scheme X mod p^n  and the category of crystalline GL(Z/p^n)-representations of the fundamental groups of the generic fibre X^0. If the time is allowed, I shall explain how our theorem ( joint with Y-H Yang) can be applied to S. Mochizuki's p-adic Teichmueller theory.