Isolated points on families of curves

Let C be an algebraic curve over Q, i.e., a 1-dimensional complex manifold defined by polynomial equations. A celebrated result of Faltings implies that all algebraic points on C come in families of bounded degree, with finitely many exceptions. These exceptions are known as isolated points. We explore how these isolated points behave in families of curves and deduce consequences for the arithmetic of elliptic curves. This talk will be suitable for a general audience.