Inter-universal Teichmüller principles in arithmetic geometry

[Lecture] Inter-universal Teichmüller principles in arithmetic geometry

Inter-universal Teichmüller theory (IUT) originates in the latest principles and techniques of anabelian geometry, in both its absolute and mono-anabelian forms, which it applies in the case of elliptic curves to Diophantine arithmetic geometry.
In this process, a new paradigm emerges, involving (a) new structures, such as certain monoids (to open up a geometry beyond the commutative algebraic geometry based on Grothendieck's ring); (b) new types of categories, such as Hodge–Theta theaters (which provide a more flexible arithmetico-geometric framework); and (c) new mechanisms, such as algorithms involving indeterminacies (which introduce an additional layer into the interpretation of commutative diagrams). Put together, these new principles encode deformations of analytic structures at the level of étale fundamental groups, thereby linking discrete invariants that are a priori unrelated -- a perspective referred to as arithmetic holomorphy philosophy.
The aim of these lectures is to provide a transversal presentation of these principles, accessible to non-experts and oriented towards their potential applications across a broader scope of arithmetic geometry. The lectures will incorporate the most recent developments in mathematical research practice. Each lecture will begin with a 30-minute introduction at the level of a general mathematical colloquium.