Ultra-high angular resolution by gravitational microlensing
  The problem of restoration of the source brightness distribution from an analysis of the stellar and AGNs microlensing light curves is investigated. In case of microlensing of stars by a point-mass lens as well as for caustic crossing events for binary lens the problem can be reduced to solution of the Fredholm integral equation of the 1st kind. Concrete form of the kernel of this equation depends on a type of the microlensing event. Assuming the circular symmetry of the stellar disk the search for radial brightness distribution can be carried out in the special compact sets of functions which correspond to the physics of the problem. These sets include the non-negative functions that are not increasing with increasing distance from the center of stellar disk and the upwards convex non-negative functions. The brightness distribution for the AGNs accretion disks is also circularly symmetric, but only in the locally co-moving frame. Therefore, the kernel of integral equation that determined the AGN microlensing light curve must take into account equally with the projection effect on picture plane the influence of relativistic effects. The search for solution of this equation can be carried out in the set of non-negative down convex functions. The results of analysis of microlensing light curves for the red giant MACHO Alert 95-30 and the A6 star MACHO 98-SMC-1 as well as the results of numerical simulations for the AGN microlensing observations are given. 
M.B. Bogdanov