Theory of equations has a long history of thousands of years
in mathematics, and, passing publication of the famous Cardano-Ferrari formulas in Italian Renaissance, Galois theory in the 19th century established a necessary and sufficient condition for an algebraic equation to have a root solution in terms of its Galois group. My research interest is a modern version of Galois theory, especially its arithmetic aspects.
In the last century, the notion of Galois group was generalized to "arithmetic fundamental group" by Grothendieck, and Belyi's discovery (of an intimate relationship between Galois groups of algebraic numbers and fundamental
groups of topological loops on hyperbolic curves) undertook a new area of "anabelian geometry". Here are important problems of controlling a series of
covers of algebraic curves and their moduli spaces, and Ihara's theory found deep arithmetic phenomena therein.
Related also to Diophantus questions on rational points, fields of definitions and the inverse Galois problem, nowadays, there frequently occur important developments as well as new unsolved problems. I investigate these topics, and hope to find new perspectives for deeper understanding of the circle of ideas.