We've seen in the seminar that while the BMO is explicitly computable, showing that it is the only obstruction can be a tricky issue. Indeed, one of the most successful strategies is to rely on a very difficult conjecture about prime values of polynomials. However, such number theoretic input is a little easier to prove "on average". We'll discuss what this means for the study of rational points and survey some recent results in this direction. Speaker(s): Nick Rome (UM)