Crew's parabolicity conjecture

I will report on recent developments of the theory of monodromy groups of convergent and overconvergent F-isocrystals. These algebraic groups have been defined by Crew using the Tannakian formalism. After quickly recalling the definitions, I will explain the main ideas of the proof of Crew's parabolicity conjecture, which provides a strong relationship between these two monodromy groups. I will continue by explaining how this comparison can be used as a bridge between "motives" and F-isocrystals/ p-divisible groups. In the end, I will also state an enhancement of the conjecture I recently obtained in a joint work with Van Hoften.