My research area is micro-local analysis, which studies properties of functions and generalized functions. In micro-local analysis we mainly investigate generalized solutions of systems of linear partial differential equations, not only on the base space but also on the phase space including variables for direction. In mathematical language the phase space is called the cotangent bundle. Via Fourier transformation, functions in x variables are transformed into those in ξ variables. In these terms micro-local analysis studies generalized functions with respect to x and ξ .
The tools of microlocal analysis vary from functional analysis and Fourier analysis to complex analysis. I mainly use the method of Algebraic analysis which was founded by Prof. Mikio Sato in RIMS, Kyoto University. More precisely, the generalized functions I study are Sato's hyperfunctions, which are cohomology classes of the sheaf of germs of holomorphic functions. Classical results in mathematical analysis such as Cauchy-Kowalevsky's theorem are reformulated in terms of the homology algebra. For this formulation, I use the language of D-modules.
The objects which I investigate are mainly solutions of hyperbolic differential equations. In the 70's, this area was well-studied by Sato-Kawai-Kashiwara when the characteristic varieties of the equations were simple. I have been interested in the case where the characteristic varieties are singular. I have also obtained a fundamental result on hyperbolic mixed problems. Recently, I have been considering the relation between asymptotic analysis and second micro-local analysis.