Explore a lecture on the algebraic counterpart of the Corlette--Simpson correspondence in p-adic geometry. Delve into the relationship between local systems and Higgs bundles on complex varieties, and discover how this transcendental concept transforms into a purely algebraic one in characteristic p. Learn about Bezrukavnikov's identification of de Rham local systems with twisted Higgs bundles, and understand how Ogus--Vologodsky recovered an untwisted correspondence. Examine the extension of this theory to smooth rigid spaces over perfectoid p-adic fields, where generalized local systems are identified with twisted Higgs bundles. Gain insights into ongoing research by Bhargav Bhatt and Mingjia Zhang, inspired by Heuer's recent work, and understand the implications for trivializing the G_m-gerbe to obtain honest Corlette--Simpson correspondences in the p-adic setting.