p-adic Hodge theory, anabelian conjecture, and arithmetic Simpson correspondence

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The project is about the crystalline comparison theorem in p-adic Hodge theory, with non-trivial coefficients and in the relative setting. During the research period, I studied in depth the work of Faltings and Scholze on integral p-adic Hodge theory. Currently I am able to formulate integral results for etale and crystalline cohomology with non-trivial coefficients. The relative version of such results shall need further investigation. The current theory of prismatic cohomology of Bhatt and Scholze seems to be the right way of formulating an integral comparison theorem with non-trivial coefficients. There has been some results in this direction, which generalizes my previous result published in 2019.