Tim Browning (IST Austria): The Hasse principle for random hypersurfaces

Take a random homogeneous polynomial of degree $d$ in $n$ variables, with rational coefficients. What is the probability that it has a non-trivial zero? When $n>d$,  Poonen and Voloch conjectured in 2002 that this should be the same probability that it is everywhere locally soluble. I shall report on joint with Pierre Le Boudec and Will Sawin towards this conjecture.