Our research areas are parallel computing, fast algorithms, computational electromagnetics, and computational sciences. We have been solving some of the most complicated problems in electromagnetics, and this translates to extremely large numerical problems. In terms of problem and matrix sizes, my research group has solved more than one billion unknowns, and this problem requires the solution of 1,000,000,000x1,000,000,000 (one billion by one billion) dense (not sparse) matrices. This is a world record for the solution of largest dense matrix problems. By solving such huge numerical problems, we gain the ability to solve very complicated physical problems. For example, one application is to use microwave/electromagnetic scattering for the purpose of medical imaging, such as diagnosing cancer tumors under the skin.