Thermoacoustic (TAT) and Photoacoustic Tomography (PAT) are examples of multiwave imaging methods allowing to combine the high imaging contrast of one wave (an electromagnetic or a photoacoustic one) with the high resolution of ultrasound. We present recent results obtained in collaboration with Gunther Uhlmann, Jianliang Qian and Hongkai Zhao on the mathematical theory behind TAT, PAT and other multiwave methods. We allows the acoustic speed to be variable, and consider the partial data case as well. We will also discuss the case of a discontinuous speed modeling brain imaging.  Numerical reconstructions will be shown as well.
Most of the progress is due to the use of microlocal methods. One of the goals of the talk is to show the usefulness of microlocal methods to solving real life problems.