21cm cosmology presents a unique opportunity to map the early Universe, bridging the vast observational gap between the Cosmic Microwave Background (CMB) last scattering surface and the high-redshift galaxies observed by JWST and ALMA. Produced by neutral hydrogen atoms in the intergalactic medium, the 21cm signal is highly sensitive to radiation from early sources of light, offering a pristine window into the signatures of metal-free Population III stars, early stellar remnants, and accreting black holes. In this talk, I will review the latest theoretical developments, existing observational constraints, and prospects for navigating this faint cosmic frontier. The formation of the first stellar generations and the subsequent emergence of X-ray binaries drove a fundamental phase transition in the early Universe. Due to immense observational challenges, the precise physical properties of these primordial light sources remain highly uncertain. The cosmological 21cm signal contains unique information about these first generations of UV and X-ray sources, as well as their feedback on the surrounding intergalactic environment. Currently, state-of-the-art radio telescopes are establishing increasingly informative upper limits, placing new constraints on early star formation and X-ray source properties. These observations are synergistic with direct information on high-redshift galaxies obtained by the JWST. In my talk, I will discuss multi-wavelength synergies which, alongside observations with the upcoming Square Kilometre Array (SKA), will open a transformative window into the cosmic dawn. This combined approach will not only allow us to characterize the properties of Pop III stars and early galaxies, but will also enable novel tests of standard cosmology and dark matter physics across an entirely unexplored redshift regime.
In many experiments in the natural sciences, we aim at gathering information on a quantity that cannot be observed directly, whereas its impact on another quantity can be measured. Generally speaking, we then have to reconstruct the cause from measurements of its impact. The evaluation of indirect data requires the solution of inverse problems. Such mathematical problems are often ill-posed, which means that even small noise in the data can lead to unacceptable errors in the reconstruction if the noise is not appropriately compensated. This depends largely on the mathematical model, which describes the relation between the quantity of interest and the measured data. The aim of this talk is to give an overview on the mathematical basics of inverse problems and discuss applications from physics, in particular from biophysics and computerized tomography.
