Anabelian Arithmetic Geometry—a New Geometry of Forms and Numbers: Inter-universal Teichmüller Theory or ‘‘Beyond Grothendieck’s Vision’’

This text presents an informal overview on how, in accordance with some deeply rooted principles of the philosophy of Alexander Grothendieck concerning the practice of mathematics, recent progress in anabelian arithmetic geometry has led to the Inter-universal Teichmüller theory (IUT) of Mochizuki Shinichi. The new geometry of monoids furnished by IUT may be understood as the result of a seminal encounter between Grothendieck’s principle of resolving the tension between the discrete and continuous realms, on the one hand, and $$p$$ -adic Hodge theory and height theory, on the other. In doing so, it opens a new research frontier that goes beyond the Grothendieck geometry of rings-schemes by providing a unifying framework for Diophantine and anabelian arithmetic geometry.