Professor Quinto will pursue problems in tomography and the mathematical analysis of Radon transforms. He will develop and refine tomographic singularity detection algorithms and apply them to electron microscopy. Professor Quinto has successfully tested his limited data tomography algorithm on electron microscope data, and he will use these results as a basis for further refinement with the goal of incorporating scattering and other effects. He will continue research on the approximate inverse in three-dimensional SONAR. Professor Quinto plans to prove uniqueness theorems for a model of synthetic aperture RADAR, research which is based on his pure mathematical work. Within pure mathematics, the principal investigator plans to prove uniqueness and support theorems for spherical Radon transforms. He and Professor Agranovsky will use these results to characterize stationary sets of solutions of the wave equation. Stationary sets, where the solution is zero for all time, are very important and difficult to characterize. He plans to prove uniqueness theorems for Radon transforms on spheres on real-analytic manifolds and use these results to begin to characterize stationary sets on rank-one symmetric spaces. He is proving Morera theorems for the distinguished boundary of polydisks, and he plans to generalize them to complex manifolds. He hopes to prove inverse continuity for an important class of local limited data problems or clarify when they do not hold.
This research encompasses both applied and pure mathematics: tomography and integral geometry. The pure research will be used to develop, understand, and justify the applied algorithms, and the applied problems will motivate much of the pure research. A computed tomography algorithm will be developed for electron microscopy and tested jointly with colleagues at the Karolinska Institute in Sweden. Our goal is to produce accurate pictures of viruses and small molecules using an electron microscope. Jointly with a colleague and an undergraduate student, he will develop an algorithm for emission tomography, a type of tomography that locates metabolic processes. He will develop pure mathematics that will show how well the algorithms work and where their limitations might occur. These pure mathematical underpinnings are required to ensure that his (or any other) algorithms are as effective as they can be. SONAR data can be modeled as averages over spheres (the spherical wave fronts of the sound wave), and Professor Quinto will jointly develop new algorithms for SONAR. This pure mathematics is intriguing in its own right. Professor Quinto will prove theorems about spherical averages which will be used to understand the wave equation (the equation that describes how sound and light behave). The wave equation describes the motion of a drum head, and he will specify which points on the drum do not move at all. Finally, he will prove theorems he will apply to complex and harmonic analysis, fields in pure mathematics.
