How many cusp form are there on SL(2), SL(n), or a more general (reductive or semisimple) linear algebraic group? Until a few years ago it was not known that there are infinitely many cusp forms on a group such as SL(n) beyond very small values of n.

Weyl's law refers to an asymptotic formula for the number of cusp forms on a given connected reductive group, in particular establishing their infinitude. I will discuss some work-in-progress, joint with Werner Mueller of University of Bonn, establishing Weyl's law with remainder terms for classical groups. Without remainder terms, this result was established, for spherical cusp forms, by Lindenstrauss and Venkatesh in a rather general setting.

Photoacoustic tomography (PAT) is an emerging soft-tissue imaging modality that has great potential for a wide range of biomedical imaging applications.  It can be viewed as a hybrid imaging modality in the sense that it utilizes an optical contrast mechanism combined with ultrasonic detection principles, thereby combining the advantages of optical and ultrasonic imaging while circumventing their primary limitations. The goal of PAT is to reconstruct the distribution of an object's absorbed optical energy density from measurements of pressure wavefields that are induced via the thermoacoustic effect.  In this talk, we review our recent advancements in practical image reconstruction approaches for PAT in heterogeneous acoustic media.  Such advancements include physics-based models of the measurement process and associated inversion methods for reconstructing images from limited data sets.  Applications of PAT to transcranial brain imaging are presented.

Advisor:  Jack Xin