"What is information?" Having this naive yet profound question in mind, I have been working mainly on noncommutative statistics, information geometry, quantum information theory, and algorithmic randomness theory.
One can regard quantum theory as a noncommutative extension of the classical probability theory. Likewise, quantum statistics is a noncommutative extension of the classical statistics. It aims at finding the best strategy for identifying an unknown quantum object from a statistical point of view, and is one of the most exciting research field in quantum information science.
Probability theory is usually regarded as a branch of analysis. Yet it is also possible to investigate the space of probability measures from a differential geometrical point of view. Information geometry deals with a pair of affine connections that are mutually dual (conjugate) with respect to a Riemannian metric on a statistical manifold. It is known that geometrical methods provide us with useful guiding principle as well as insightful intuition in classical statistics. I am interested in extending information geometrical structure to the quantum domain, admitting an operational interpretation.
I am also delving into algorithmic and game-theoretic randomness from an information geometrical point of view.
Someday I wish to reformulate thermal/statistical physics in terms of algorithmic information theory.