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 Teichmüller spaces, Kleinian groups, Complex Dynamics, and fractals, as well. Tanaka Lab studies various combinatorial objects, including codes and combinatorial designs. Their underlying spaces often have strong symmetry/regularity, and the representation theory of the semisimple algebras naturally associated with these spaces is the main tool in our analysis of combinatorial substructures.