The first inverse problem we will consider is whether we can determine the structure of spacetime when we see light coming from many point sources varying in time? We can also observe gravitational waves.We will define the concept of light observation set to model this problem and show that the light observation set determines the topology, differential structure and conformal type of the metric of the set of light sources.
The second question we discuss is related to the Cosmic Microwave Background (CMB). This is the thermal radiation remnant from the Big Bang The inverse problem we study is the determination of early gravitational perturbations from CMB measurements.
This is joint work with Y. Kurylev, M. Lassas and P. Hintz for the first part and Matti Lassas, Lauri Oksanen and P. Stefanov for the second part.