This proposal presents a two part research program that creates a synergistic interaction between mathematics and astrophysics. Program 1 aims at developing a mathematical theory to determine the probability distributions of Gaussian curvatures at critical points of random time-delay functions. This research program opens up new mathematical directions that bridge singularity theory with probability theory (e.g., zeros of random polynomials and limit theorems in Rayleigh-Levy statistics). In addition, Program 1 has direct applications to understanding the nature of dark matter in galactic halos via the flux ratio anomalies in the lensing signatures of galaxies. Program 2 seeks to extend the classification of light curves and image centroids of sources near caustics in thin-screen, weak-field microlensing to the strong gravitational field of a Kerr spacetime. This involves studying the global topology, geometry, and singular theoretic structure of light caustic surfaces in a spacetime. Program 2 points to new mathematical issues relating the geometry of light caustic surfaces to topological invariants of the matter singularities of gravitational lenses. This program may also yield tests of Einstein's General Theory of Relativity, especially as it pertains to the massive black hole believed to lie in the nucleus of our galaxy.
The dark matter in galactic halos and the black hole in the center of our galaxy are two central and pressing topics in astrophysics. These issues are directly impacted by the powerful mathematical methods of geometric analysis because gravitational lensing is simultaneously a unique tool for probing the nature of dark matter distributions and black holes, and a theory that is built on geometric, analytical, and probabilistic concepts. The dark matter study requires developing a mathematical theory that allows one to differentiate generic dark matter lensing signatures from features that are specific to the particular choice simple lens model used for dark matter. The first part of the proposal deals with formulating such a mathematical theory. This work would open up new mathematical directions that bridge the geometric aspects of singularity theory with several areas in probability theory (Rayleigh-Levy statistics, zeros of random polynomials, etc.). The second part of the proposal focuses on one of the fundamental predictions of Einstein's General Theory of Relativity, namely,the existence of black holes. Though a black hole cannot be seen directly, its extremely strong gravitational field warps the spacetime about the black hole causing light rays that managed to get through the region to be bent with impressively large angles (e.g., rays can loop around a black hole numerous times before arriving at the observer). The probing of such extremes of gravity would be a critical test of our understanding of the nature of space and time. The proposal explores mathematically the lensing signatures of the strong gravitational field due to the massive black hole generally believed to be at the center of our galaxy. This should draw upon techniques from differential geometry and singularity theory, and create synergies between these mathematical topics and the physics of black holes.