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.
This is contained in papers 1) and 2).
The second question we discuss is related to the Cosmic Microwave Background (CMB). This is the thermal radiation remnant from the Big Bang (discovered by Penzias and Wilson 1964). It is considered as a primary source of information regarding the early universe. For example, the EGS (Ehlers-Geren-Sachs,
1968) theorem roughly states that the isotropy of observed CMB implies the isotropy of the universe. The inverse problem we study is the determination of early gravitational perturbations from CMB measurements. For This we study the light ray transform. This is contained in papers 3) and 4)/
1) Y. Kurylev, M. Lassas and G. Uhlmann, Inverse problems for
Lorentzian manifolds and non-linear hyperbolic equations. Invent.
Math. 212 (2018), no. 3, 781–857.
2) P. Hintz and G. Uhlmann, Reconstruction of Lorentzian manifolds from boundary light observation sets. Int. Math. Res. Not. IMRN 2019, no. 22, 6949–6987.
3) M. Lassas, L. Oksanen, P. Stefanov and G. Uhlmann, On the inverse problem of finding cosmic strings and other topological defects. Comm.
Math. Phys. 357 (2018), no. 2, 569–595.
4) A. Vasy and Y. Wang, On the light ray transform of wave equation solutions.
Comm. Math. Phys. 384 (2021), no. 1, 503–532.
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