Abstract: The Brauer-Severi variety a $x^2 + b y^2 = z^2$ has a rational point if and only if the cup product of cohomology classes with $F_2$ coefficients associated to $a$ and $b$ vanish. The cup product is the order-$2$ Massey product. Higher Massey products give further structure to Galois cohomology, and more generally, they measure information carried in a differential graded algebra which can be lost on passing to the associated cohomology ring. For example, the cohomology of the Borromean rings is isomorphic to that of three unlinked circles, but non-trivial Massey products of elements of $H^1$ detect the more complicated structure of the Borromean rings. Analogues of this example exist in Galois cohomology due to work of Morishita, Vogel, and others. This talk will first introduce Massey products and some relationships with non-abelian cohomology. We will then show that $b x^2 =$ $(y_1^2 - a y_2^2$ $+ c y_3^2 - ac y_4^2)^2 - c(2 y_1 y_3 - 2 a y_2 y_4)^2$ is a splitting variety for the triple Massey product $\langle a,b,c \rangle$ with $F_2$ coefficients, and that this variety satisfies the Hasse principle. It follows that all triple Massey products over global fields vanish when they are defined. Jan Minac and Nguyen Duy Tan have extended this result to all $\mathbb{F}_p$ and with $p=2$ to all fields of characteristic different from $2$. The method discussed in the talk could produce splitting varieties for higher order Massey products. This is joint work with Michael Hopkins.