Abstract Proposal: DMS-9734586 Principal Investigator: Arlie Petters This project centers on two mathematical programs dealing with the optical caustics created by gravitational fields. One program investigates the magnification cross sections of caustic-curve networks due to generic multiplane weak-field gravitational lens systems. The other program studies the links between the global geometry of the optical caustic surfaces in a spacetime (e.g., along our past light cone or about rotating black holes) and the singularities of the associated spacetime metric. These issues point to a new and promising direction in the mathematical theory of optical caustics. Furthermore, their rigorous treatment may also lead to a deeper understanding of the flux enhancement of gravitationally lensed quasars and their overdensity around foreground galaxies, and rapid X-ray variability in active galactic nuclei. Equally important, such investigations may reveal new and powerful ways of understanding and organizing fundamental physical concepts in gravitational lensing, and can provide a theoretical check on results from numerical simulations. Gravitational lensing is the deflection of light from a distant source (e.g., quasar, extended galaxy) by an intervening matter distribution (e.g., galaxy, cluster of galaxies). When foreground galaxies magnify background quasars, it causes an ``amplification bias'' in certain quasar samples. This has led to a mathematically difficult and physically important problem in gravitational lensing: Determine the probability that a light source is magnified greater than some specified amount. A rigorous study of this problem directly bears on the the issue of ``magnification bias.'' Another important problem is to investigate gravitational lensing by dynamical lenses (e.g., rotating black holes). This is physically relevant since most galaxies seem to have black holes at their centers. Moreover, gravitational lensing by a rotatin g black hole may account for rapid X-ray variability in active galactic nuclei. The aforementioned problems will lead me to interact with astrophysicists and astronomers for physics discussions and latest observations, computer scientists for sophisticated numerical codes running on high-performance computers, and mathematicians for issues in singularity theory and differential geometry. Also, implicit with this general aim is isolating and studying the underlying mathematical structures invoked by gravitational lensing. Once determined, these structures can then be made available as potential mathematical tools for other physical systems (possibly shock formation in fluids) that may call upon them.
