Modular curves, modular forms and Hecke operators: old and new

Ogg’s celebrated conjecture can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic curves) exist if and only if there is a good geometric reason for them to exist. Ogg’s mathematical viewpoint has inspired an increasingly broad array of results and conjectures. B. Mazur’s talk will discuss this and recent related work.