Computerized tomography plays a central role in biomedical imaging. Since the invention of computer-aided tomography in the 1960s, numerous imaging modalities have been introduced and became indispensable for diagnostic instruments in biology and medicine. Recently, the quest for more sensitive and more reliable techniques has led to such promising coupled-physics modalities as Thermoacoustic and Photoacoustic Tomography (TAT and PAT), and Magnetoacoustoelectric Tomography (MAET). These imaging methods combine high resolution of ultrasound with the sensitivity of electromagnetic waves to optical absorption and conductivity of the tissues. Sharp abnormalities in the latter physical parameters are good markers of breast cancer, thrombosis, ischemia, and other medical conditions. Thus, these new techniques overcome limitations of classical tomography, and deliver otherwise unavailable, potentially life-saving diagnostic information - at a lesser cost and with less harm to a patient. The images in these modalities are obtained by complex mathematical procedures, rather than through direct acquisition. The mathematics of these methods is, mostly, at very early stages of development. The investigator and his collaborators work to resolve the central theoretical problems and to develop efficient numerical techniques for PAT, TAT, MAET and other hybrid techniques. A graduate student and a postdoc are playing a significant role in the project, gaining exposure to the exciting area at the junction of exact sciences, medicine, and biology. The results will be disseminated through publications in high quality research journals, presentations at national and international conferences, and series of lectures at various major venues. The mathematics underlying and enabling such modalities as PAT, TAT, MAET and several novel techniques based on Compton scattering, contains a number of challenging open problems, important from both the theoretical and applied points of view. The investigator and his collaborators aim to gain a theoretical understanding and to develop algorithmic foundations for these modalities. In particular, they work on (1) deriving exact inversion formulas for the problem of TAT/PAT reconstruction from the sets of data reduced both in time and in space; (2) developing efficient reconstruction algorithms for different MAET data acquisition schemes, including MAET with loss of low frequencies, and MAET of objects with anisotropic conductivity; (3) devising efficient numerical techniques for congregating strongly over-determined Compton data sets, and processing them using attenuation compensation techniques previously developed for Single Photon Emission Computed Tomography. In addition, the newly developed theoretical and algorithmic tools will be used to process real MAET data obtained by the PI in experimental MAET research done jointly with the researchers from the Medical Imaging Department at the University of Arizona. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.