Algebraic language theory, Galois theory and class field theory

Takeo Uramoto – 23 June 2021

In this talk I introduce my recent work on interplay between automata theory and number theory. In automata theory, it has been well known that there are tight connections between regular languages, finite monoids, and finite automata; and via such connections, one can sometimes solve decision problems on logical / combinatorial properties of regular languages by reducing them to purely algebraic problems on finite monoids. This phenomena are somewhat analogous to those known in galois theory: one can check whether solutions of a polynomial can be represented by arithmetic operations and roots by checking whether the galois group of the minimal splitting field of the polynomial is soluble. In fact, this analogy is not just an analogy but can be justified via categorical axiomatizations of the classical result in automata theory, yielding a unification of classical ideas in automata theory and galois theory. Through this unification, I also introduce some ideas of automata theory shed new light on classical (explicit) class field theory too. If time allows, I will discuss some future perspectives.