The research subject of Sugawa Lab is mainly Complex Analysis. Even if the data and/or functions are described in terms of real variables, hidden structures may emerge when dealing with them as complex variables. For instance, in the classical problems of moments concerning a sequence of real numbers, the power series formed by the sequence (the generating function) gives us many useful visions to tackle the problems. In such a case, Complex Analysis plays an important role. We are studying analytic functions from the geometric viewpoint to provide new interpretations to classical results. Moreover, we are interested in quasiconformal mappings, which have recently found many applications in image processing and brain mapping. With the help of computers together with the above knowledge, we are studying modern topics such as Teichmller spaces, Kleinian groups, Complex Dynamics, and fractals, as well.

In the Matsubara-Heo Lab, we study differential equations from the perspectives of special functions and algebraic analysis. Our central research interest lies in hypergeometric functions, which are "universal special functions" as they appear in various mathematical contexts. More recently, we have also been exploring research topics and methods derived from related fields such as algebraic statistics and quantum field theory. Terms like period integrals, algebraic de Rham cohomology groups, intersection theory, D-modules, multivariate discriminants, and convex polytopes may seem daunting at first glance. However, all these concepts serve as crucial tools for discovering new insights under the broad theme of the theory of differential equations.