G-functions, Periods, and irrationality

In 1978, Apery proved that zeta(3) was irrational. Perhaps unexpectedly, his methods have
proven exceptionally hard to generalize. In recent work with Dimitrov and Tang, we have found a new framework with which to study these ideas, with a number of applications both to modular forms and irrationality. Our aim will be to give a gentle overview of our ideas, as well as highlight a number of basic open questions about G-functions which Riemann would understand.