Abstract：In algebraic geometry, families of smooth varieties give rise to numerous invariants, among which are the notable Betti-local systems and the Kodaira-Spencer maps—graded Higgs bundles. Direct summands of these local systems and graded Higgs bundles are called motivic.  Our understanding of motivic local systems is currently limited, with only a few known cases. Noteworthy examples include the rigid rank-2 local systems and the cohomologically rigid rank-3 local systems. Nevertheless, the broader question of determining motivic conditions for local systems and Higgs bundles in a more comprehensive context remains a challenging and ongoing pursuit in algebraic geometry.

In this short course, we will introduce a p-adic method that addresses the determination of motivicity for specific types of local systems and Higgs bundles. The proof incorporates various important elements, including the theory of Fontaine-Faltings modules, Higgs-de Rham flow, $p$-to-$\ell$ companion, Drinfeld's theory on the Langlands correspondence, Grothendieck-Messing-Kato deformation theory, and a numeric Simpson correspondence established by Hongjie Yu. As a consequence, we obtain infinitely many algebraic solutions to the Painleve VI equation. This short course is based on a joint work with Kang Zuo.