Even though diagonals of multivariate rational functions have been studied from various viewpoints, they still remain quite mysterious objects. An example for this is the widely open conjecture by Christol which characterizes diagonals inside the class of all D-finite functions. In 2012 Bostan, Boukraa, Christol, Hassani, and Maillard created a list with 116 potential counter examples for this conjecture. As of today, using new kinds of identities involving diagonals and hypergeometric functions, 40 of these examples were resolved by the starting work of Abdelaziz, Koutschan and Maillard and the generalization by Bostan and the speaker.
In the talk I will explain how the key identities were found and proven, indicate their various implications, and finally mention limitations and possible extensions. The talk is based on joint work with A.~Bostan.