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.
- Superdimensional Metamaterial Resonators by Allan Greenleaf, Henrik Kettunen, Yaroslav Kurylev, Matti Lassas, Gunther Uhlmann Posted Sept. 11, 2014; submitted. (ArXiv preprint)
- On necklaces inside thin subsets of R^d by Allan Greenleaf, Alex Iosevich, and Malabika Pramanik Posted Sept. 9, 2014; submitted. (ArXiv preprint)
- A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral by Allan Greenleaf, Alex Iosevich, Bochen Liu and Eyvindur Palsson Posted June 15, 2013; revised August 26, 2013, and April 10, 2014. Revista Mathematica Ibereoamerica, to appear. (ArXiv preprint)
- Multilinear generalized Radon transforms and point configurations by Loukas Grafakos, Allan Greenleaf, Alex Iosevich and Eyvindur Palsson Posted April 19, 2012; Forum Mathematicum, published online Sept. 2013, DOI: 10.1515/forum-2013-0128. (ArXiv preprint)
- On volumes determined by subsets of Euclidean space by Allan Greenleaf, Alex Iosevich and Mihalis Mourgoglou Posted October 31, 2011; Forum mathematicum , 27, no. 1, (2015),635 - 646. Published online , June 19, 2013 (subscription req.). (ArXiv preprint)
- Cloaked electromagnetic, acoustic and quantum amplifiers via transformation optics by Allan Greenleaf, Yaroslav Kurylev, Matti Lassas, Ulf Leonhardt and Gunther Uhlmann Proc. National Acad. Sci., published online before print June 14, 2012, doi: 10.1073/pnas.1116864109; print: 26 June 2012, v. 109, no. 26, 10169-10174.
- On triangles determined by subsets of the Euclidean plane, the associated bilinear operators and applications to discrete geometry by Allan Greenleaf and Alex Iosevich Posted September 14, 2010; revised Oct. 19, 2011. Analysis & PDE 5-2 (2012), 397-409. DOI 10.2140/apde.2012.5.397. (ArXiv preprint)
- A multi-dimensional resolution of singularities with applications to analysis by Tristan Collins, Allan Greenleaf and Malabika Pramanik Posted July 4, 2010; American Journal of Mathematics, 135, no. 5, (2013), 1179-1252. Published online, DOI: 10.1353/ajm.2013.0042 (subscription req.). (ArXiv preprint)
- Fourier integral operators with open umbrellas and seismic inversion for cusp caustics by Raluca Felea and Allan Greenleaf Posted January 27, 2010; Mathematical Research Letters , 17 (2010), 867-886. (ArXiv preprint)
- Cloaking a Sensor via Transformation Optics by Allan Greenleaf, Yaroslav Kurylev, Matti Lassas and Gunther Uhlmann Posted December 9, 2009; revised Oct. 31, 2010. Phys. Rev. E 83, 016603 (2011) . (ArXiv preprint)