In the 1960s, Grothendieck dreamt that algebraic varieties can be linearized in a universal way, leading to his philosophy of motives. Subsequent ideas of many mathematicians (especially Beilinson and Deligne) led to a beautiful conjectural framework surrounding the notion of a motive. In the last decade, thanks to the discovery of perfectoid geometry and subsequent developments, some aspects of this framework have also been realized unconditionally in the context of p-adic motives on p-adic varieties. In these lectures, I will survey some of this landscape, with an emphasis on the concrete applications that have guided the theoretical developments.
This is the second part of two talks, the first of which may be found here.
This video is part of Harvard University‘s conference Current Developments in Mathematics 2023.
