Title: A positive proportion of quartic binary forms does not represent 1.

Abstract. I will discuss an explicit construction of many equations of the shape F(x , y) = 1 which have no solutions in integers x, y, where F(x , y) is a quartic form with integer coefficients. In this recent work, in order to construct a dense subset of forms that do not represent 1, the quartic forms are ordered by the two generators of their rings of invariants. In a previous joint work with Manjul Bhargava, we showed a similar result, but we ordered forms by their naive heights.