Abstract: Adamczewski and Bell proved in the 2013 the Loxton - van der Poorten conjecture. It says the following. Let f be a Laurent power series (with complex coefficients) and let \sigma_p be the operator substituting x^p for x in f. Suppose that f  satisfies a homogenous polynomial equation in the operator \sigma_p with coefficients which are rational functions, and a similar equation in the operator \sigma_q where p and q are multiplicatively independent natural numbers. Then f is a rational function. The original proof was pretty complicated. In a recent preprint Schafke and Singer gave a conceptual striking new proof, as well as new proofs of other results of the same nature that existed in the literature. The goal of the seminar is to explain their proof. We shall also explain an application to the theory of automata, saying that the only sets of natural numbers which are simultaneously p-automatic and q-automatic are finite unions of arithmetic progressions (Cobham's theorem). The two talks will be elementary. Only background needed is Galois theory and complex analysis.