Prof. Greenleaf's research interests are in harmonic analysis and microlocal analysis, with applications to integral geometry
and inverse problems. In recent years, he has been particularly interested in estimates for oscillatory integral and Fourier integral operators with degenerate phase functions. These arise in looking at solutions to certain partial differential equations, and from averaging operators associated with families of curves or lines in n-space. The latter include X-ray transforms which provide the mathematical underpinnings of CAT scanning. Recently, Prof. Greenleaf has also been interested in multiplicative properties of
Fourier integral distributions. Controlling these allows one to obtain uniqueness and reconstruction in various inverse problems,
such as determining a potential function from the backscattering data ( a subset of the scattering kernel of the associated wave equation)
or from the Cauchy data of the associated time-independent Schrödinger equation.
More recently, Prof. Greenleaf, together with Matti Lassas of the Helsinki University of Technology, Yaroslav Kurylev of University College, London, and Gunther Uhlmann of the University of Washington, have been using insight gained from the study of inverse problems to give a rigorous foundation and introduce new constructions in the burgeoning field of "cloaking", or invisibility from observation by electromagnetic waves.
Prof. Greenleaf's research is supported in part by National Science Foundation grants.