New intersections between class field theory and algebraic language theory

Reference No.	20210006
Type/Category	Grant for Young Researchers-Short-term Joint Research
Title of Research Project	New intersections between class field theory and algebraic language theory
Principal Investigator	Takeo Uramoto（Institute of Mathematics for Industry, Kyushu University・assistant professor）
Research Period	August 9,2021. ～ August 13,2021. January 27,2022. ～ January 29,2022.
Keyword(s) of Research Fields	algebraic language theory, semi-galois category, class field theory, finite automata
Abstract for Research Report	The purpose of this research is two-hold: first, to provide new insights on classical problems in classical theory of computation; and second, to provides new toolkits for number theory. This research has its background in algebraic language theory, a branch in formal language theory, which has been developed in relation with classifications of computational hierarchies of formal languages, and particularly concerns its semigroup-theoretic methodology for classifications of regular languages. Recently, the representative of this research proved a duality theorem between semi-galois categories and profinite monoids, unifying algebraic language theory and galois theory, and observed interesting relationships between the classical ideas in algebraic language theory and class field theory, in particular, complex multiplication. The representative of this research regards this observation as something fundamental and promising; thus, in this research project, we shall further develop this direction, having the following perspectives in mind: first, from the standpoint of number theory, the above phenomenon may give new insights on classical class field theory, and as its sequel, may provide a new formulation of non-abelian class field theory; second, from the standpoint of computability theory, the above phenomenon gives new problems to classical computability theory, arising from number theory. Moreover, the above phenomenon, which can relate classical discrete models of computation (finite automata) with arithmetic geometric objects (lambda rings) in a certain precise category-theoretic sense, may provide new methodology for classical computability theory where our conventional methods have been only combinatorial (rather than geometric). Recalling the fact that appropriate geometric frameworks have been behind foundations of physics, this phenomenon, if developed more, may contribute to the foundation of computability theory. For the time being, it is not easy to foresee how this development can be influential for applications, but the representative of this research has several directions in mind. Among them, the most important one is a contribution to the foundation of computational complexity theory. As mentioned above, the root of this research is in the classification theory of computational or logical complexity hierarchies of regular languages, which has been remarkable and a good model for our subject on classifying computational hierarchies. In fact, there is a recent trend in this field to extend this theory for regular language to higher classes of languages, including the works of the representative of this research himself. As seen from the fact that P vs NP problem is still open, the problem of classifying computational hierarchies is in general very difficult and tends to be ad hoc, while this problem is fundamental for application areas such as cryptography. In view of this, it would be reasonable, even from the standpoint of applications, to try to extend the classification theory of regular languages to higher classes. Our research itself concerns only the theory of classification of regular languages but by developing its foundations from new perspective of number theory, we expect that we gain some new (geometry-origin) insights towards extensions of this classical theory of classifications of regular languages.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Takeo Uramoto（Institute of Mathematics for Industry, Kyushu University・assistant professor） Yasuhiro Ishitsuka（Institute of Mathematics for Industry, Kyushu University・assistant professor）
Adviser	Takuro Abe (IMI, Kyushu University / Professor)